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TRANSCRIPT
Design Project
Master in Mechanical Engineering
Design and analysis of the Engine Reservoir of the BuriedOcean Battery
Author:Nuria Masclans Serrat S4162935
Supervisors:1st supervisor: Marijn van Rooij
2nd supervisor: Antonis Vakis
December 6, 2020
Preface
This Design Project report is the result of a three months work conducted by the author in collab-oration with Ocean Grazer Company and University of Groningen. The project aims to developand to perform a mechanical analysis of a preliminary design for the Engine Reservoir of the OceanBattery, a project in progress of Ocean Grazer Company.
I would like to thank my supervisors Marijn van Rooij and Proj. dr. Antonis Vakis for giving methe opportunity to make a contribution in such a promising and sustainable project, and to guideme along all the process. I would also like to express my gratitude to Prof. dr. Wout Prince,whose bright ideas where always of a big help.
Finally, I would like to give very special thanks to my family, who I miss very much.
Nuria Masclans SerratDecember 5, 2020
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Abstract
Nowadays, the sector of energy production is suffering a significant increase on renewable energy.Therefore, renewable methods for energy storage must be developed accordingly. Ocean Grazercompany believes on a renewable, clean and reliable way of storing energy produced by renewablepower sources. The company is currently developing the so-call Ocean Battery, a system of flexibleand rigid reservoirs based on the principle of storing the produced energy by renewable power plantsas potential energy. It is expected to be located on a sea or lake bottom, with a Rigid Reservoirburied on the soil and a Flexible Reservoir laying on the soil surface. Although some research hasbeen developed on the design of the Rigid and Flexible Reservoirs, Ocean Grazer has detecteda lack of knowledge regarding the third component of the battery: the Engine Reservoir. TheEngine Reservoir is the interconnecting element between the Rigid and Flexible Reservoirs, andit contains the hydro-power components required for energy storage and production. Therefore,the project focuses on providing a first design of the Engine Reservoir. In order to do so, therequirement of the component are identified and the main hydro-power components characterised.Additionally, a study on the medium interaction with the structure is performed, and the relevantloads and momentum are identified and quantified through Matlab and COMSOL models. Finally,the mechanical study is performed, and the design evaluated.
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Contents
1 Introduction 8
2 Problem Definition 92.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Design Scope 93.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Goal Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Methodology 114.1 Hydro-power components characterisation and Design description . . . . . . . . . . 114.2 Engine Reservoir interactions characterisation and Mechanical analysis . . . . . . . 11
4.2.1 COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.2 CYPECAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Design criteria and requirements 125.1 Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Design Input Data 146.0.1 Geological and Geotechnical data . . . . . . . . . . . . . . . . . . . . . . . . 14
7 Hydro-power components 147.1 PHES Main Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Hydro-power components characterisation . . . . . . . . . . . . . . . . . . . . . . . 17
7.2.1 Characterising the reversible pump/turbine . . . . . . . . . . . . . . . . . . 177.2.2 Characterising the water conveyance system . . . . . . . . . . . . . . . . . . 227.2.3 Characterising the reservoirs capacity . . . . . . . . . . . . . . . . . . . . . 237.2.4 Characterising the pump flow rate . . . . . . . . . . . . . . . . . . . . . . . 24
8 Elaboration of the design geometry and material 268.1 Characterising the Engine Reservoir walls: thickness and reinforcement . . . . . . 268.2 Design I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Design II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9 Literature research: Engine Reservoir interaction loads 309.1 Stability conditions: uplift and driving . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.1.1 Structure vertical deadweight . . . . . . . . . . . . . . . . . . . . . . . . . . 309.1.2 Geotechnical load carrying capacity . . . . . . . . . . . . . . . . . . . . . . 319.1.3 Stability conditions expressions . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.2 Seabed → Structure interactions: Earth Pressures . . . . . . . . . . . . . . . . . . 349.3 Fluid → Structure interactions: Hydrostatic and hydrodynamic conditions . . . . 35
9.3.1 Fluid-Structure interaction: Hydrostatic loads . . . . . . . . . . . . . . . . . 369.3.2 Fluid-Structure interaction: Hydrodynamic or fluid flow loads . . . . . . . . 36
9.4 Rigid Reservoirs and Flexible Bladders Connection Loads . . . . . . . . . . . . . . 37
10 Model setup 3910.1 Model I. Stability conditions on structure uplift and driving and soil load carrying
capacity (Matlab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2 Model II. Fluid → Porous Seabed Interaction (Comsol) . . . . . . . . . . . . . . . 3910.3 Model III. Fluid → Structure Interaction (Comsol) . . . . . . . . . . . . . . . . . . 4010.4 Model IV. Porous Seabed → Structure Interactions (Matlab) . . . . . . . . . . . . 4210.5 Model V: Integrated Model (Comsol) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10.5.1 Governing equations of the fluid sub-model . . . . . . . . . . . . . . . . . . 4310.5.2 Governing equations of the seabed sub-model . . . . . . . . . . . . . . . . . 4410.5.3 Governing equations of the structure sub-model . . . . . . . . . . . . . . . . 45
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10.5.4 PDE Coefficients for seabed sub-model . . . . . . . . . . . . . . . . . . . . . 4610.5.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
10.6 Model VI - Integrated model II (Comsol) . . . . . . . . . . . . . . . . . . . . . . . 4710.7 Model VII - Reinforced structure (Cypecad) . . . . . . . . . . . . . . . . . . . . . . 48
11 Simulation Results 5111.1 Results Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.2 Results Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.3 Results Model III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11.3.1 Fluid-Structure Interaction: Hydrostatic loads . . . . . . . . . . . . . . . . 5311.3.2 Fluid-Structure Interaction: Hydrodynamic loads . . . . . . . . . . . . . . . 53
11.4 Results Model IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.5 Results Model VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.6 Results Model VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12 Discussion 66
13 Conclusion 67
14 Recommendations and Limitations 68
A Metocean Data 71
B Hydrodynamic environmental modelling 74B.0.1 Currents modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74B.0.2 Waves modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C Velocity profile fitting for normal and extreme state conditions 77
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List of Figures
1 Scheme of the Ocean Battery current designs: the ’Buried’ Design and the ’SuctionAnchor-based’ Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 PHES configurations: (a) quaternary, (b) ternary, (c) binary . . . . . . . . . . . . . 163 Schematic relationship between the power (grey isocurbes, in MW), the capacity
(Q, in m3/s), the total differential head (TDH, in m and logarithmic axis) and theturbine type (delimited regions) of a pump/turbine. The red cross corresponds tothe chosen pump/turbine for the design. The original image has been taken fromSulzer public portfolio [33], and modified afterwards. . . . . . . . . . . . . . . . . . 18
4 Relationship between Head (in meters) and Specific Speed (Ns or ns) for diverseturbine types. The red cross corresponds to the chosen pump/turbine for the design.The original image has been taken from [30], and modified afterwards. . . . . . . . 19
5 Turbine and pump/turbine weight versus the impeller diameter. Image obtainedfrom [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Calculated turbine and penstock design parameters for the studied range of valuesof the design head (Hd). The studies parameters are (a) Capacity or dischargeflow, (b) Specific speed, (c) Rotational speed, (d) impeller/runner diameter and (e)penstock diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Estimation of (a) Reservoirs Capacity and (b) Pumping flow rate for the studiedvalue ranges of power production and turbine design head. . . . . . . . . . . . . . . 24
8 Lateral view of the initial shape and structure of the Engine Reservoir. The OceanBattery main systems are itemised using letters: the Flexible Bladders (A), theEngine Reservoir (B) and the Rigid Reservoirs (C). The main components relevantfor the requirements satisfaction and for the mechanical analysis are numbered (1-9). 27
9 Schematic lateral view of some initial design variables of the Engine Reservoir. . . 2810 Top view of the Machine Hall, with the schematic distribution of the main En-
gine Reservoir in colours. The hydraulic and electrical components detailed are:pump/turbine (yellow), motor/generator (orange), penstocks (dark blue), water con-veyance system except penstock (light blue), shaft elevator (green) and additionalarea for transformers, control system and other other ancillary rooms (gridded re-gion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11 Scheme on the interactions between the Engine Reservoir, the other Ocean Batterymain components (Flexible Bladders, Rigid Reservoirs) and the medium domains indirect contact to the Engine Reservoir (fluid and seabed soil). . . . . . . . . . . . . 31
12 Geometry and free and porous media flow boundary conditions of model I. Boundarynumbering in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
13 Schematic view of the cross-section of a river bank. Specified the governing equationsof the fluid flow for each soil region []. . . . . . . . . . . . . . . . . . . . . . . . . . 41
14 Geometry and boundary conditions of Model III. . . . . . . . . . . . . . . . . . . . 4115 Geometry and boundary conditions of Model V . . . . . . . . . . . . . . . . . . . . 4316 Surface boundary loads as pressures applied on the structure external surface due
to the one-way interactions of the fluid and seabed to the structure. . . . . . . . . 4817 Cypecad 3D representation of the Engine Reservoir model. . . . . . . . . . . . . . 4918 Visualisation of Cypecad floors. Height defined with the origin in the seabed surface. 5019 Parametric study on ER embedded depth of static ER deadweights, buoyancy and
soil capacities. Soil considered: Silica Sand, cohesionless. . . . . . . . . . . . . . . . 5120 Parametric study on ER embedded depth of stability safety factors against uplift
and driving. Soil considered: Silica Sand, cohesionless. . . . . . . . . . . . . . . . . 5221 Parametric study on ER embedded depth of stability safety factor against uplift
(SFuplift) for multiple cohesive (continuous curves) and cohesionless (dotted curves)soils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
22 Effective vertical stress and lateral earth stress at-rest for Kaolin Clay (black) andSilica Sand (green). Pore water pressure (blue) for both types of soil. . . . . . . . . 54
23 Soil displacement along soil depth for different values of Young Modulus for soil type(a) Kaolin Clay and (b) Silica Sand. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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24 Plane representation of the fluid velocity field. Decomposition of the fluid velocityfield in x,y,z-components in yz-plane in x=100m (structure centre in x-direction). . 55
25 Plane representation of the fluid velocity on significant xy and xz-planes. Velocitymagnitude (m/s) surface and velocity field streamlines. . . . . . . . . . . . . . . . . 56
26 Total stress distribution on the structure surface on interaction with the fluid. De-composition of the total stress in x,y,z-direction. . . . . . . . . . . . . . . . . . . . 56
27 Pressure difference between the simulated hydrodynamic pressure (p) and the hy-drostatic pressure (p*) calculated analytically (Eq. 45). The hydrodynamic pres-sure profile (p(z)) is taken from the simulation for x = 97.3m and y = 0.01 andz ∈ (0, 35), which corresponds to a vertical line along the fluid domain and that iscontained in the surface of the Face A. . . . . . . . . . . . . . . . . . . . . . . . . . 57
28 Lateral earth pressure at-rest as a function of depth below the seabed surface (σr(z)).Used dseabed = 35m and soil parameters of Kaolin Clay (Table 2). . . . . . . . . . . 58
29 First, second and third principal stresses on the structure external surface due toone-way interaction of fluid and seabed to the Engine Reservoir. Stress magnitudesas surface plot, stress directions as arrows. . . . . . . . . . . . . . . . . . . . . . . . 59
30 Von Mises stress (N/m2) for the external (a) and internal (b) wall surfaces of theEngine Reservoir structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
31 Total displacement magnitude (m) and scaled deformation of the Engine Reservoirexternal wall surface using pure concrete walls. . . . . . . . . . . . . . . . . . . . . 60
32 3D representation of the deformation suffered on the Engine Reservoir walls, slabsand pillars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
33 Displacement in z-direction and moments in x,y-direction on the upper concrete slabof the Machines Hall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
34 Total and x,y-direction shear stresses on the upper concrete slab of the Machines Hall. 6335 Amount of reinforcement steel required (cm2/m) for the upper concrete slab of the
Machines Hall. Reinforcement considered: superior and inferior reinforcement forboth x,y-directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
36 Slab Reinforcement of the project floors P2-P5. Floor P1 is not added because itrepresents the foundations, it does not contain an horizontal slab. . . . . . . . . . . 65
37 Decomposition of the steel mass (a) and the concrete volume (b) used in each floorand for each structural component. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
38 Dataset covered area (delimited with a purple line) and colour-map of the meanwater depth (land areas in grey). Included Dutch offshore wind farms (delimitedby a black line): Borssele (1), Hollandse Kust zuid (2), Hollandse Kust noord (3),Hollandse Kust west (4), Ten Noorden van de Waddeneilanden (5) and IJmuidenVer (6) [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
39 Regions of applicability of different wave theories as a function of the dimensionlesswave steepness (H/gT 2
app) and the dimensionless relative depth (d/gT 2). The wavetheories considered are Stream Function, Stokes V and Linear Wave Theory [1] . . 76
40 Measured current velocity profiles (orange circles, data from Tab. 13) and predictedvelocity profiles (continue curves). The velocity profiles have been predicted usingthe nonlinear model from Eq. 114,115 (blue) and from Eq. 116 (red) . . . . . . . . 78
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List of Tables
1 Design criteria and requirements for the Engine Reservoir design . . . . . . . . . . 132 Soil parameters for cohesive soils: Kaolin clay, Nkossa clay and Qiantang river silt
clay [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Soil parameters for cohesionless soils: Silica sand, Statoil sand, Redhill 110 sand
and Luce Bay sand [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Rule thumb of weight of steel reinforcement per cubic meter of concrete of different
concrete building elements [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Initial values of the Engine Reservoir design variables. . . . . . . . . . . . . . . . . 286 Terzagui’s shape factors [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Soil parameters used on Model II [4] . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Boundary conditions of the physics in Model II, using the boundary nomenclature
detailed in Fig. 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Model VI porous seabed parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 4310 CYPECAD floors description. *Surface loads only applied on [12 x 8] floor region. 4911 Caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512 Project summary in terms of concrete surface (m2), concrete volume (m3) and re-
inforcement steel bars mass (kg). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213 Current velocity data for TNW normal sea state (’Mean’) and extreme sea state
(’Max’). Current depth average and current velocity profile data for specific per-centages of the water column [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14 Significant wave height and associated peak wave period for TNW normal sea state.The significant wave height (HS,NSS) is discretised by 0.5m steps, and the rangeof peak wave period (TP,NSS) associated to each wave height is presented by thevalues of 5%, 50% and 95% of the data [5]. . . . . . . . . . . . . . . . . . . . . . . 73
15 TNW metocean extreme values with a 50-years return period [2]. . . . . . . . . . . 7316 Computation of dimensionless wave steepness (H) and relative depth (d) for the case
of normal sea states. Results for the lowest, intermediate and largest significant waveheight and their associated peak wave periods (Tab. 14). . . . . . . . . . . . . . . . 76
17 Fitted coefficients and fitting errors of the nonlinear models regarding the meancurrent velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
18 Fitted coefficients and fitting errors of the nonlinear models regarding the maximumcurrent velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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1 Introduction
The word is suffering a rising need of sustainable, clean and affordable energy. Specially in the fieldof energy storage, there is a clear predominance of mineral based batteries (such as lithium-basedbatteries), which contain materials that need to be mined and furthermore they are very difficultto recycle. Ocean Grazer identified this deficiency, and is currently developing a Ocean Batterycapable of storing energy in a completely renewable way [7].
The project of the Ocean Battery stated on 2014 within a research group on the University ofGroningen. Ocean Grazer company was founded few years after, and nowadays is currently testingin the field the first prototype of the Ocean Battery at small scale. Based on storing energy throughpotential energy, the Ocean Battery is compatible with all types of power plants, including hybridpower plants. IT must be of high versability to adapt to every implementation requirements, androbust to withstand de extreme pressure conditions of the sea or lake bottom.
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2 Problem Definition
As introduced in the Preface, Ocean Grazer company has no preliminary design for the EngineReservoir of the Ocean Battery. Furthermore, there is a gap on knowledge on the system require-ments and on the design parameters of the hydro-power components within the Engine Reservoir.Furthermore, there is no clear description of the mechanical interactions of the Engine Reservoirwith the environment and with the other components of the Ocean Battery.
2.1 Problem Statement
From the problem definition introduction, it is clear that Ocean Grazer company requires a cleardesign methodology for the Engine Reservoir, to be followed for the first Engine Reservoir designbut also flexible and well-detailed so that the design can be attractively optimised by reviewingthe design steps. Accordingly, the problem statement is as follows:
Ocean Grazer company has a gap of knowledge on the Engine Reservoir requirements, on the mainhydro-power machinery that must be contained inside and on the nature of the interactions withother Ocean Battery components and the environment. Consequently, the company is not able todevelop a first design of the Engine Reservoir.
3 Design Scope
The problem identified in the previous section is of high dimensions. Therefore, a clear scope andboundaries of the design must be set, so that it is performed successfully and with the requiredquality.
3.1 System Description
Ocean Grazer company is currently working in two very different designs: the ’Buried’ Designand the ’Suction Anchor-based’ Design (Fig. 1). Although it is clear that its geometry is clearlydifferent, they base on the same fundamentals: the energy storage through potential energy. Thedesign project focuses on the design of the Engine Reservoir or Engine Room of the ’Buried’ System.
The basic principle of the system is the following. There is a close circuit of fluid; the fluid can bestored in the Rigid Reservoir and in the Flexible Reservoir or Flexible Bladder, and both reservoirsare connected through a water conveyance system. The Rigid Reservoir is buried under the sea orlake soil (refered from here as seabed for the sake of briefty), and it contains both water and airin atomspheric pressure. That is to say, the Rigid Reservoir stored water is at atmospheric pres-sure. Contrarilly, the Flexible Reservoir is located on the seabed surface, and its flexible surface orbladder deforms without offering much resistance. Therefore, the Flexible Reservoir stored fluidfeels the sea/lake hydrostatic pressure applied on the Bladder surfaces; the fluid is at the seabedhydrostatic pressure. Consequently, the fluid stored on the Flexible Reservoir has significantlymore energy than the fluid stored on the Rigid Reservoir due to the height difference betweenreservoirs and due to the pressure increase described previously.
It has already been said that the Ocean Battery is to be implemented as a battery for renewablepower plants. At a certain moment, it can be of interest of the power plant to store producedenergy on the Ocean Battery. Then, the Ocean Battery stores this additional energy in the form ofpotential energy by pumping the fluid from the Rigid Reservoir (low energy fluid) to the FlexibleBladder (high energy fluid). Contrarily, the grid might demand energy at a certain moment, andthen the Ocean Battery discharges by dropping the fluid from the Flexible Bladder to the RigidReservoir through a turbine. Therefore, the Engine Reservoir must contain all the hydro-power andelectric components to store energy (battery charging) and to deliver energy (battery discharging).
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Figure 1: Scheme of the Ocean Battery current designs: the ’Buried’ Design and the ’SuctionAnchor-based’ Design
3.2 Goal Statement
The goal statement is formulated from the problem statement (Section 2.1) and the system descrip-tion (Section 3.1). From a general problem context of lack of a Engine Reservoir comprehensivedesign, the following goals have been defined:
• Identify the Engine Room or Reservoir design requirements
• Make a preliminary design of the hydropower components
• Clearly identify the main interactions between the Engine Reservoir and sea/lake, the seabedsoil and the Rigid and Flexible Reservoirs.
• Elaborate a method for calculating the main loads derived from the identified interactions.
• Elaborate a first design and analyse its mechanical performance.
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4 Methodology
As observable on the previous sections, the project encompass a high variety regarding the liter-ature research and field of study. There is a important research work to be done on hydro-powercomponents, which will conclude on a approximate layout of the components inside the EngineReservoir. On the other side, there is research to be done on the behaviour of the seabed, the fluidand the solid structure, and the main loads that drive its interactions. This workload must becomplemented with simulations and qualitative results, and it will conclude with the mechanicalanalysis of the design proposed in the first part of the project, and its evaluation. Accordingly,both the workload and the report are structure in two parts: ’Hydro-power components characteri-sation and Design description’ and ’Engine Reservoir interactions characterisation and Mechanicalanalysis’.
4.1 Hydro-power components characterisation and Design description
The hydro-power components characterisation is the result of extensive literature research on thefield of Pumped Storage Hydroelectricity or PHES. Some approximate calculations will be doneas a function of the design variable dburied, which refers to the Engine Reservoir base depth withrespect to the seabed or soil surface (or the equivalent design variable Hd, introduced furtherin the report). It has been chosen to maintain dburied as a variable for a significant part of thework in order to make the design method as general and flexible as possible, and to stress thesignificant effect of the buried depth in many stages of the design. Finally, some approximationsand simplifications are made for the development of the first design of the Engine Reservoir bytaking into account the system requirements and characterised hydro-power components.
4.2 Engine Reservoir interactions characterisation and Mechanical anal-ysis
Firstly, a research work is done in the field of the interactions between the Engine Reservoir andthe other Ocean Battery components (Rigid Reservoir and Flexible Bladder) and between theEngine Reservoir and the environment (Fluid and Soil domains). Following to the literature re-search, some simulation and analytical models are developed in order to quantify the interactions,and identify which of them are significant for the mechanical behaviour of the structure. Lastly, acomprehensive model of the Engine Reservoir that includes the significant interactions is simulatedand evaluated.
Matlab software is used for some analytical calculations. When simulation tools are required, twoFinite Element Method based softwares are used: COMSOL Multiphysics and CYPECAD.
4.2.1 COMSOL Multiphysics
COMSOL Multiphysics is a Finite Element Method solver that can simulate a bast quantity ofphysics phenomena. It is specialised on the following physics: electromagnetism, structural me-chanics, acoustics, fluid flow, heat transfer and chemical engineering [?]. Through a user friendlyinterface, the physics governing equations are implemented through the description of the geometry,the material domains and the boundary conditions. It is used to model the fluid and structure basicbehaviour, but the lack of the proper licences has difficulted the simulation of the soil behaviour.
4.2.2 CYPECAD
CYPECAD software is a Finite Element Method solver specialised on steel and concred reinforcedstructures. It carries an analysis and design of the structures, and it allows the application ofhorizontal and vertical forces on the project design [12]. It has been implemented to account forthe steel reinforcement and in order to obtain a more complete mechanical analysis that COMSOLcould not provide.
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5 Design criteria and requirements
In order to develop the Engine Reservoir design, the design criteria and requirements must beclearly identified. They are defined in the following subsections.
5.1 Design criteria
The design considerations are guided by the philosophy of the so-called Limit State Design [23].The design has to satisfy the following criteria:
• Ultimate Limit State (ULS): It requires the estimation of the maximum loads due to allpossible design load cases, and comparing them with the load absortion capacity of thestructure. The main loads taken into account are the overturning moment and the verticaland lateral loads. This consideration requires the computation of the ultimate moment andthe lateral and axial load capacities. Therefore, ULS will provide an estimation of the therequired minimum structure dimensions and shape. The ULS inputs are given by the locationcharacteristics, such as the wave and current data. The earthquake and ice loads of extremeenvironmental conditions are out of the scope.
• Fatigue Limit State (FLS) and long term deformation: It requires the prediction of the fatiguelife and the long term cyclic loading of the structure.
• Robustness and ease of installation: It determines whether the structure can be installedwith the required redundancy.
• Functionality: It determines the purpose and actions of the Engine Reservoir, such as con-taining the necessary machinery for a Pumped-Storage Hydropower station and allowing theflow between flexible and rigid reservoirs. Functionality requirements are defined by thestakeholder Ocean Grazer.
• Compatibility: The design of the Engine Reservoir must be compatible with the design of theburied rigid reservoir/s and the flexible bladder/s that constitute the Ocean Battery system.The compatibility requirements are given by the stakeholder Ocean Grazer, and include themultiple connection between the Ocean Battery main components (Engine Reservoir, FlexibleBladder and Rigid Reservoir) and the required depth of each component.
The design requirement are summarised in Table 1 and related to design criteria described above.
12
5.2
Desi
gn
Requir
em
ents
Req
uir
emen
tid
.C
ateg
ory
Des
crip
tion
Lim
it
R1
R1.
AU
LS
Th
elo
adca
rryin
gca
paci
tyof
the
En
gin
eR
oom
has
toex
ceed
the
maxim
um
pos
sib
lelo
ads.
Th
eco
nsi
der
edlo
ad
sare
the
late
ral
an
dve
rtic
al
load
san
dth
eov
ertu
rnin
gm
om
ent.
Fh,ULS<Fh,ER
Fv,ULS<Fv,ER
MULS<MER
R1.
BU
LS
Th
eE
ngi
ne
Room
yie
ldst
ren
gth
has
toex
ceed
the
stru
ctu
rem
axim
um
stre
ss.
σm<f yk
R2
R3.
AS
LS
Th
ein
itia
lan
dac
cum
ula
ted
defl
ecti
on
hav
eto
be
less
than
0.5
m.
ρ0<
0.5m
,ρacc<
0.5m
R3.
BS
LS
Th
ein
itia
lan
dac
cum
ula
ted
tilt
hav
eto
be
less
than
0.5º
and
0.2
5º
resp
ecti
vely
.θ 0<
0.5º
,θ acc<
0.25º
R3
R3.
AF
un
ctio
nal
ity
Th
eE
ngi
ne
Room
mu
stb
eis
ola
ted
an
dco
nta
inair
at
atm
osp
her
icp
ress
ure
R3.
BF
un
ctio
nal
ity
Th
eE
ngi
ne
Room
mu
stco
nta
inth
em
ach
iner
yfo
ra
pu
mp
ed-s
tora
ge
hyd
rop
ower
.
R3.
CF
un
ctio
nal
ity
Th
eE
ngi
ne
Room
mach
iner
ym
ust
ensu
rea
pro
du
ctio
nca
paci
tyof
1to
2M
Ww
ith
red
un
dan
cy1M
W<P<
2MW
R3.
DF
un
ctio
nal
ity
Th
eO
cean
Bat
tery
mu
sth
ave
ad
isch
arg
eti
me
of
1/2
hou
ran
da
char
gin
gti
me
of
2-4
hou
rs.
Tdischarge
=0.
5hTcharge
=2h
R3.
EF
un
ctio
nal
ity
Th
eE
ngi
ne
Room
mu
stfo
rese
efu
ture
des
ign
mod
ifica
tion
s,su
chas
anin
crea
seof
the
pro
du
ctio
nca
paci
tyan
dof
the
nu
mb
erof
rese
rvoir
s.R
3.F
Fun
ctio
nal
ity
Th
eE
ngi
ne
Room
mu
stall
owm
ain
ten
an
ceand
rep
lace
men
top
erati
on
s
R4
R4.
AC
omp
atib
ilit
yT
he
En
gin
eR
oom
mu
stb
eco
nn
ecte
dto
sin
gle
/m
ult
iple
flex
ible
bla
dd
ers
thro
ugh
wat
erfl
owin
pu
ts/ou
tpu
ts.
#co
nn
ecti
on
s:nFB
z FB,connection>z F
B,bottom
R4.
BC
omp
atib
ilit
yT
he
En
gin
eR
oom
mu
stb
eco
nn
ecte
dto
the
sin
gle
/m
ult
iple
rigid
rese
rvoir
sth
rou
ghw
ater
flow
inp
uts
/ou
tpu
ts.
#co
nn
ecti
on
s:nRR
R4.
CC
omp
atib
ilit
yT
he
En
gin
eR
oom
’sp
um
pm
ust
be
loca
ted
at
alo
wes
th
eigh
tth
an
the
low
est
wat
erle
vel
ofth
eri
gid
rese
rvoir
/s.
z PT<z R
R,bottom
Tab
le1:
Des
ign
crit
eria
an
dre
qu
irem
ents
for
the
En
gin
eR
eser
voir
des
ign
13
6 Design Input Data
Previous to the hydro-power components characterisation and the interaction loads analysis, theinput data of the system must be defined. It refers to the environmental conditions as well asthe system location under the water level. Initially, the Engine Reservoir had to be located inthe sea environment, therefore data was collected for a specific location in the Northern sea wherethere is a wind-power plant. As the system has been finally decided to be located on a lake, themeteocean data (currents, waves and wind related data) has been considered of less importance forthe project. Then, the collected data has been moved to Appendix A, to be consulted for furtherprojects of Ocean Grazer in the sea environment.
6.0.1 Geological and Geotechnical data
The water depth considered is d = 35m, which corresponds to the mean water depth of the TNWwind farm zone [2]. For initial studies, the seabed is considered flat and of depth d. It is of high im-portance to perform a complete soil study for a project such as the Engine Room, as soil-structureinteraction is highly dependent on soil properties. For instance, the skin friction is modelled dif-ferently for cohesive and non-cohesive soils (Section 9.1.2). Due to the fact that the Ocean Grazerhas not yet decide on the exact location of the Engine Reservoir, there is no accurate data of thesoil that interacts with the structure. Therefore, several types of soil are considered (Table 2,3),and its performance compared.
The maximum buried depth of dburied,max = 50m is provided by the stakeholder Ocean Grazer.This parameter refers to the maximum depth that the Engine Reservoir and the Rigid Reservoircan be buried below the mudline. The bigger the buried depth, the smaller flow rate is required forthe same turbine power production. Nevertheless, the stress felt from the embedded structure wallsand caused by the soil with depth. Therefore, the choice of the design variable dburied ∈ (0, 50)m isa compromise between the discharge flow (or generation power) and the strength of the structure.
7 Hydro-power components
The hydro-power components necessaries for satisfying the Engine Reservoir requirements of pro-duction and storage of energy are the same as the Pumped Hydropower Storage plants (PHES)[22]. They are described in Section 7.1 and characterised for the Engine Reservoir requirements inSection 7.2
7.1 PHES Main Components
The analogy with PHES is that the Flexible Bladder works as the ’upper reservoir’, and the RigidReservoir as the ’lower reservoir’. The main components are briefly described below.
Headrace tunnelDrives the fluid from the upper reservoir or Flexible Bladder to the penstock.
PenstockConduit for water flow between the upper reservoir (Flexible Bladders) and the turbine. Pipesare used for above-groud penstocks, and shaft or tunnels for below-ground cases. The typical flowvelocity range is 1-5 m/s and typical diameter of 5-10 m. The cross-sectional area design is a tradeoff between cost and efficiency for a given flow rate Q: a large diameter provides a sower flow withlower losses but higher construction cost.
Surge tanksSurge tanks of hydraulic bypass capacitors located on the penstock or on the tailrace. Their func-tion is to absorb high pressure transients that might occur during startup or mode changeover.
14
Variable name SymbolValue
UnitKaolin clay Nkossa clay
Qiantangriver silt clay
Unified Soil Classification USC OH CH OH [-]Effective soil weight γ′ 6500 6000 8823 [N/m3]Internal angle of friction φ 26 19 36.5 [º]Adhesion factor α 0.5 0.45 0.5 [-]Soil cohesion c 10 11 10 [kPa]Mudline undrained shear strength cu,mud 4.0 5.0 6.0 [kPa]Mudline undrained shear strength rate mcu 1.50 1.67 1.35 [kPa/m]Lateral earth pressure coefficient at-rest K0 0.562 0.674 0.405 [-]Overburden bearing capacity Nq 14.210 6.701 50.351 [-]Self-weight bearing capacity Nγ 12.624 4.693 61.390 [-]Cohesion bearing capacity Nc 27.085 16.558 66.695 [-]
Table 2: Soil parameters for cohesive soils: Kaolin clay, Nkossa clay and Qiantang river silt clay[18].
Variable name SymbolValue
UnitSilica sand Statoil sand
Redhill110 sand
Luce Baysand
Relative density - loose very dense loose very dense [-]Effective soil weight γ′ 5993 8500 7820 10300 [N/m3]Internal angle of friction φ 36 45 36 45 [º]Limit unit end bearing qult,lim 2.9 12.0 2.9 12.0 [MPa]Limit unit skin friction flim 67.0 114.8 67.0 114.8 [KPa]Lateral earth pressure coefficient at-rest K0 0.412 0.293 0.412 0.293 [-]Overburden bearing capacity Nq 47.156 173.285 47.156 173.285 [-]Self-weight bearing capacity Nγ 56.655 271.717 56.655 271.717 [-]
Table 3: Soil parameters for cohesionless soils: Silica sand, Statoil sand, Redhill 110 sand and LuceBay sand [18].
15
Draft tube:Vertical conduit that drives the fluid from the turbine to the Tailrace tunnel.
Tailrace tunnelConduit for water flow between the pump and the lower reservoir (Rigid Reservoirs). Typically,they have lower pressure and flow rate than the penstocks, and therefore larger diameter. It isrequired a downward slope from the lower reservoir to the pump in order to create a pump inlethead that prevents cavitation in pumping operation.
Power house or Engine ReservoirContains the pump, turbines, motor and generators. It is typically located bellow the lowerreservoir level so that there is the required inlet head for the pump. Three configurationts arepossible for the power machines (Fig. 2), and its eligibility depends on the design requirements forefficiency, dimensions and cost:
• Quaternary set: on e pump, turbine, motor and generator. The pump and turbine arecompletely decoupled; the pump is driven by the motor and the generator is driven bythe turbine. High efficiency is achieved as both the pump and turbine operate a its bestefficiency conditions, but the equipment and infrastructure costs are notably higher than theother configurations.
• Ternary set: one turbine, one pump and one reversible motor/generator, all connected to thesame shaft. Typically, the turbine is rigidly coupled to the reversible motor/generator, whilethe pump is coupled to the shared shaft with a clutch. High efficiencies are achieved, as thepump and turbine are optimised individually. Turbine and pump can operate simultaneously,and the turbine can also be used during pump startup to reduce the changeover time.
• Binary sets: one reversible pump/turbine (or pump working as a turbine) and one reversiblemotor/generator. It is the configuration with the lowest costs (simplified hydraulic pathwaysand less hydraulic and elecrical machines). Contrarily, it provides a lower efficiency thanthe other configurations as the reversible pump/turbine design is obtained as a compromisebetween the performance of the pump and the turbine. Additionally, the changeover time islarger than for other sets because the shaft and motor/generator have to change its rotationaldirection.
Nowadays, it is commonly used binary configuration on PHES applications due to the significantreduction of costs. Nevertheless, ternary configuration is used instead of binary configurationwhen the head and flow requirements exceed the operational ranges of the available reversiblepump/turbines. Additionally, ternary configurations are also preferred for large PHES due to itshigher efficiency.
(a) (b) (c)
Figure 2: PHES configurations: (a) quaternary, (b) ternary, (c) binary
16
7.2 Hydro-power components characterisation
Requirement R5.B states that the Engine Reservoir must contain the main elements that charac-terise the powerhouse of a conventional pumped-storage hydropower station defined in the previoussection. The Engine Reservoir machinery must allow the Ocean Battery to act as a battery (en-ergy storage) and as a power station (energy generation). In order to do so, it must include thefollowing elements: turbines, pumps, generators, motors, transformers, control system and ancil-lary rooms (electrical rooms, drainage gallery, cooling systems). Moreover, it is possible that theEngine Reservoir contains some elements for water conveyance, such as the penstocks and multiplesteel gates and valves.
The scope of the design project only considers the Engine Reservoir elements that have a non-negligible influence on the mechanical study of the structure and the fulfilment of the require-ments. For the equilibrium state and loads analysis, an approximated machinery weight is used,which is obtained from the selection of the heavy machinery (turbines, pumps, generators, mo-tors, transformers). For the eigenfrequency analysis, a turbine and pump selection (following therequirement R5.C) is necessary to determine the operating frequency and to compare it with thestructure natural frequency. The flow rate at the Engine Reservoir connections is extracted fromthe turbine selection.
According to the requirements and the project description, the structure must be fully underwaterand half-buried on the seabed and it has to allow for an environment of atmospheric pressure inside.Therefore, the design intents to reduce as much as possible the dimensions of the Engine Reser-voir, while still allowing redundancy and maintenance operations. Taking these requirements intoconsideration, reversible machine sets are eligible for the design, as they save both equipment andcivil costs. In particular, it is required the use of reversible pump/turbines and reversible electricmotor/generators. The reversible pump/turbines can work as a pump or as a turbine depending onthe rotation direction. Every pump/turbine set is coupled to a reversible electric motor/generator,that works as a motor for the pump mode of the pump/turbine (storage operation of the OceanBattery cycle) and as a generator for the turbine mode (electricity production operation of theOcean Battery cycle).
7.2.1 Characterising the reversible pump/turbine
The main characteristics of a reversible pump/turbine are the output production, the best effi-ciency head, the turbine and pump operating head ranges, the impeller or runner diameter andthe rotational speed. This characteristics will be approximated for the capacity production andredundancy requirements using the equations and relationships from [25].
The turbine head depends on the depth of the flexible bladder/s and the rigid reservoir/s. From theprevious section 6.0.1, it is obtained a mudline depth of zmudline = 35m (??) and Rigid Reservoirand Engine Reservoir depth below the mudline of dburied ∈ (0, 50)m. Accordingly, the total depthof the Rigid Reservoir is zRR = zmudline+dburied ∈ (35, 85)m. For the case of the Flexible Bladder,it is clear that it must be located underwater and at the maximum depth possible (in order tomaximise its energy storage capacity). From the previous argument, it is assumed the limit caseof the flexible bladder located just above the mudline, at the maximum depth of zmudline. Theturbine best efficiency head is defined as the difference of hydraulic pressure between the flexiblebladder and the flexible reservoir, in units of height of the water column. For the system consid-ered, the pressure difference is caused by the water column above the flexible bladder (of heightzmudline) and the height difference between the flexible bladder and the rigid reservoir (dburied).Therefore, the best efficiency head is given by HBE = zmudline + dburied ∈ (35, 85)m. It should benoted that the effective head decreases as the rigid reservoir fills, but this consideration has beenneglected for simplicity and because of the lack of a definitive design of the Rigid Reservoir.
The requirements state that the Engine Reservoir machinery must ensure a production capacityof 1 to 2MW with redundancy. It is decided to include 4 reversible pump/turbines of 0.5MW in
17
the design, so that the efficiency and redundancy criteria are satisfied. From Figure 3 and usingthe data of 0.5MW of power production capacity and HBE ∈ (35, 85)m of total differential head,the chosen reversible pump/turbine is of Francis type. For minimum HBE = 35m it is required acapacity of Qt1 = 1.58m3/s (turbine t1 in Figure 3) and for maximum HBE = 85m it is obtained0.1 < Qt2 < 1m3/s (turbine t2 in Figure 3).
101.2101.4101.6101.8
100.2100.4100.6100.8
t2
t1
Figure 3: Schematic relationship between the power (grey isocurbes, in MW), the capacity (Q, inm3/s), the total differential head (TDH, in m and logarithmic axis) and the turbine type (delimitedregions) of a pump/turbine. The red cross corresponds to the chosen pump/turbine for the design.The original image has been taken from Sulzer public portfolio [33], and modified afterwards.
The velocity ratio φ1 establishes the relationship between the rotational speed n (revolutions/minor r/min), the discharge diameter of the impeller D (m) and the best efficiency head HBE (m)developed by the pump or absorbed by the turbine as:
φ1 =πnD
(60[s/min])(2gHBE)1/2=
(11.8 · 10−3[min/m1/2])nD
H1/2BE
(metric) (1)
From Eq. 1, and choosing the design head Hd as the best efficiency head, it is direct to obtain theexpression of the impeller/runner diameter:
D =φ1H
1/2d
(11.8 · 10−3[min/m1/2])n, (2)
Additionally, the available power (P ) from the stored water in the flexible bladders is calculatedfrom:
P = ρQtgHdηt, (3)
where P = 0.5MW is the available power at full gate capacity, ρ = 1000kg/m3 is the water density,Qt is the water flow (m3/s) of the reversible pump/turbine inlet when it operates in turbine mode(flow from the flexible bladder system), g = 9.81m/s2 is the gravity acceleration, Hd ∈ (35, 85)mis the designed head where best (peak) efficiency is obtained and ηt = 0.80 is the turbine efficiency.Finally, the turbine design specific speed (nst) is given by:
nst =nP
1/2KW
H5/4d
, (4)
18
t1
t2
Figure 4: Relationship between Head (in meters) and Specific Speed (Ns or ns) for diverse turbinetypes. The red cross corresponds to the chosen pump/turbine for the design. The original imagehas been taken from [30], and modified afterwards.
where n is the rotational speed in revolutions/min and PKW = 500KW is the turbine full-gatecapacity in KW.
Substituting Eq. 3 for the chosen turbine parameters and the allowed range of design head values,it is obtained the water flow Qt inversely proportional to the design head as Qt = 63.7105/Hd.The result is shown in Fig. 6(a), and the limit values of the turbine capacity are Qmax = Qt1(Hd =35m) = 1.82m3/s and Qmin = Qt2(Hd = 85m) = 0.75m3/s. The capacity values match with theapproximated data from Fig.3.
For the calculation of the rotational speed (n) using Eq. 4, it is first required to get an approximateexpression of the turbine design specific speed for the case of the chosen Francis turbine. FromFig. 4, it is clear that the decimal logarithms of design head and specific speed are related linearlyas:
log10 ns = m log10Hd + b, (5)
where the fitting coefficients m = −0.5857 and b = 3.4721 are calculated using approximate datafrom Fig. 4. The resulting specific speed for the range of values of design head is shown in Fig.6(b). Substituting Eq. 4 for the studied range of Hd, it is obtained the required rotational speedfor a specific value of Hd. The result is shown in Fig. 6(c), and the limit values for the rotationalspeed are nmin = nt1(Hd = 35m) = 1407.5rev/min and nmax = nt2(Hd = 85m) = 2537.7rev/min.
Similarly, the impeller/runner diameter expression (Eq. 2) is evaluated using the calculated val-ues of rotational speed for every design head and the approximation φ1 = 1 for a pump/turbineworking at the best effective head (for both turbine and pump operation modes). The resultingdiameter as a function of design head is shown in Fig. 6(d). The figure shows small variation ofthe diameter for the whole range of allowed design head, with Dmax = Dt1(Hd = 35m) = 0.308mand Dmax = Dt2(Hd = 85m) = 0.356m. This result implies that the turbine size (and weight) willbe the same for all possible buried depths, but the capacity or flow rate and the rotation speedwill change with the buried depth. For the weight calculation below it is used the mean value
19
Figure 5: Turbine and pump/turbine weight versus the impeller diameter. Image obtained from[25]
20
(a) Capacity or Discharge Flow (Q) as a function of Design Head (Hd)
(b) Specific Speed (ns) as a function of Design Head (Hd)
(c) Rotational Speed (n) as a function of Design Head (Hd)
(d) Impeller/Runner Diameter (D) as a function of Design Head (Hd)
(e) Penstock Diameter (Dp) as a function of Design Head (Hd)
Figure 6: Calculated turbine and penstock design parameters for the studied range of values of thedesign head (Hd). The studies parameters are (a) Capacity or discharge flow, (b) Specific speed,(c) Rotational speed, (d) impeller/runner diameter and (e) penstock diameter.
21
D = 0.332m.
The weight approximate calculation is based on the data from Figure 5. It shows the relationshipbetween the impeller diameter D and the pump and pump/turbine weight Wpt. Unfortunately,it has not been possible to find weight data for the small dimensions and production capacity ofthe design, as the existing pumped-storage facilities have notably bigger production capacity andconsequently bigger impeller diameter for the reversible pump/turbines. Consequently, in order todetermine the weight of the chosen reversible pump/turbine, it has been used the approximationthat the relationship shown in Figure 5 for impeller diameter D ∈ (0.5, 15)m is also true for thesmaller diameter D = 0.332m = 1.089ft. Then, an approximate calculation of the reversiblepump/turbine weight is obtained from the exponential regression of the figure (a continuous linein a log-log representation). Using the data from the figure, the linear regression for the log-logplot at the smallest range of impeller diameter (continuous line) can be written as:
log10Wpt = m log10D + b, (6)
where the fitting coefficients m = 2.193 and b = 3.047 are calculated using Fig. 5 approximatedata.Substituting m and b values in 6 for the calculated impeller/runner diameter of D = 0.33m =1.08feet, it is obtained the reversible pump/turbine weight Wpt =' 1319pounds = 598kg ≈ 600kg.
7.2.2 Characterising the water conveyance system
The water conveyance system is composed by the following elements: water intakes (working asinlets and outlets), supply works, penstocks and tailraces [11]. The water intakes must be at thelowest point of the flexible reservoirs, and the water is conducted from the water intakes to thepenstocks through the supply works or headraces. The penstocks are the conduits or pipes thatbring the water from the intakes or the supply works to the reversible turbine-pump overcoming thehead difference. They are usually made of steel, and they must ensure high pressure on the insidein case there are sudden load increases or decreases. Finally, the tailrace connects the reversiblepump/turbine to the rigid reservoir.
The penstocks diameter is of special interest for the Engine Reservoir design, as it should be takeninto consideration for the estimation of the Engine Reservoir dimensions. Although the flexiblebladder intake and supply work is out of the scope of the project, it is worth mentioning thattypically every reversible pump/turbine has its own penstock, but the multiple penstocks may ormay not share the same water intake or supply channel. The penstock diameter (Dp) is directlyrelated to the penstock rated discharge (QD), which coincides with the turbine mode inlet flow(Qt) calculated in Eq. 3. The optimum diameter for small hydropower projects and as a functionof the rated discharge was developed by Warnick et al. (1984) as:
Dp = 0.72√QD (7)
Substituting QD = Qt in the previous expression and using the relationship between the turbineflow rate with the design head (Eq. 3, the penstock diameter is calculated as:
Dp = 0.72
√P
ρgHdηt∝ H−0.5d (8)
The optimum Dp for the available range of Hd is shown in Fig. 6(e); it decays with Hd as H−0.5d .Then, the maximum value of Qp,max = 0.97m is obtained for minimum design head Hd = 35m,and the maximum Qp,min = 0.62m is obtained for maximum Hd = 85m. From this result and therequirement of being able to install up to 4 reversible pump/turbines of 500KW, it is obtained anadditional requirement for the Engine Reservoir: The Engine Reservoir has to allow the installa-tion of up to 4 steel penstocks of diameter Dp ∈ (Dp,min, Dp,min) which will depend on the burieddepth of the structure.
22
Additionally, an estimate weight computation can be made for more accuracy in loads calculation.It is used that the penstock is made of steel, as it is common for small hydropower projects. Inaddition, it is used a totally vertical penstock to minimise its length and the Engine Reservoirdimensions. Then, the penstock length corresponds to the height difference between the flexiblebladder intake and the pump/turbine bladders. Using the assumption that the flexible bladderintake must be near the seabed, the penstock length is approximately Lp ' dburied ∈ (0, 50)m.The maximum penstock weight (Wpenstock, [N]) is found when it is full of water:
Wpenstock =
[πD2
p
4ρw +
π
4((Dp + e)2 −D2
p)ρsteel
]gdburied (9)
The penstock wall thickness (e, in m) is calculated as [6]:
e =ptotalDp
2σsteel(10)
where p (Pa) is the total internal pressure within the conduit, Dp (m) is the outside diameter ofthe penstock, and σsteel is the allowable tensile stress of the steel. The total internal pressure iscaused by the static pressure (p) and the water hammer (ph):
ptotal = p+ ph (11)
The maximum value of the static pressure p is found at the bottom of the penstock, and is givenby p = ρwg(zmudline + dburied) ∈ (0.34, 8.33)MPa. Additionally, the water hammer appears whenthe turbine vale is closed, the flow stops abruptly and the dynamic energy is converted into elasticenergy of pressure waves that travel back and forth the penstock (they are damped out due tofriction). The velocity of the pressure wave in the penstock (cp) can be approximated as ph = 0.20pfor practical purposes [6]. Then, the internal pressure range for the range of admitted dburied isptotal = 1.2p ∈ (0.41, 10.0)MPa [19].
For normal conditions (earthquakes and penstock filling and draining not considered), the rec-ommended allowable tensile stress is the smallest value between 2/3 the minimum yield stressand 1/3 the minimum tensile strength of the steel. Carbon steel (JIS SS400) is found to beused in the penstocks from hydroelectric power plants [16]. With σy = 235MPa as minimumyield stress and σt = 400MPa as minimum tensile strength [32], the penstock allowable stress isσsteel = min(2/3 · σy, 1/3 · σt) = 133.33MPa.
Penstock thickness (e) is calculated substituting the previous values in Eq. 10. For the limit caseof Engine Reservoir and Rigid Reservoir buried dburied = 50m below the mudline, it is obtained amaximum penstock thickness of e = 23.25mm.
The maximum penstock weight is approximately calculated for the maximum value of the designvariables: Dp = 0.97m and e = 0.023m. Substituting these values and carbon steel density ofρsteel = 7850kg/m3 in Eq. 9, it is obtained that the penstock weight (Wpenstock(dburied)) will notexceed the limit of Wpenstock(dburied) ≤ 9.98 · 103dburied < 104dburied [N ].
7.2.3 Characterising the reservoirs capacity
Reservoirs capacity refers to the maximum water capacity of the system of (multiple) FlexibleBladders and (multiple) Rigid Reservoirs. For simplicity, it has been assumed the same capacityfor both the system of Flexible Bladders and Rigid Reservoirs (CFB = CRR = Creservoirs). Thewater capacity of the reservoirs (Rigid Reservoirs and Flexible Bladders) depends on the powerproduction (1 < P < 2MW , from Requirement R5.C), the discharge time (Tdischarge = 0.5h, fromRequirement R5.D), the turbines design head (35m < Hd < 85m, from Section 7.2.1) and theturbines discharge flow for a certain power production and turbine design head (Qt1 = 1.82m3/sfor P = 0.5MW and Hd = 35m and Qt2 = 0.75m3/s for P = 0.5MW and Hd = 85m, fromSection 7.2.1).
23
(a) Reservoirs Capacity, Creservoirs(m3) (b) Pumping flow rate, Qp (m3/s)
Figure 7: Estimation of (a) Reservoirs Capacity and (b) Pumping flow rate for the studied valueranges of power production and turbine design head.
From the turbine power expression (Eq.3) and using the relationship between energy and power,it is obtained:
Edischarge =
∫ Tdischarge
0
Pdt =
∫ Tdischarge
0
ρwCreservoirsgH(t)ηt(H(t))dt ≈ (12)
≈ ρwCreservoirsgHdηt (13)
Secondly, the total energy produced by the complete discharge of the flexible reservoirs can beapproximately calculated as:
Edischarge ≈ PTdischarge, (14)
where Creservoirs is the total capacity of the system of flexible reservoirs in units of volume ofwater (m3 of water). This expression uses the approximation that the turbine efficiency (ηt(H))and turbine head (H(t)) are constant through all discharge operation, and that the turbine headis equal to the design head (Hd). The approximation has been used due to the lack of informationabout the number of reservoirs and its shape and allowable water levels, which would provide theevolution of H(t) during discharge, and the its corresponding ηt(H). Combining Eq. 13 and 14 itis obtained the expression for the total water capacity of the (multiple) flexible bladders:
Creservoirs =PTdischargeρwgHdηt
(15)
where Tdischarge, ρw, g and ηt are known parameters. It is clear to see that that the volumecapacity is directly proportional to the power production, and inversely proportional to the designhead. Fig. 7(a) visualises the result of the Creservoirs calculation using Eq. 15 and range of valuesP ∈ (1, 2)MW and Hd ∈ (35, 85) aforementioned.
7.2.4 Characterising the pump flow rate
The pump characteristics derive from the reservoirs volume and desired pumping period. Then,the pumping period (Tcharge) refers to the total time required to pump the water from the systemof Rigid Reservoirs in its full capacity to the system of Flexible Bladders, initially in its minimumcapacity. Assuming a constant flow rate for the whole pumping process for simplicity, it is direct toobtain that the pumping flow rate (Qp) depends on the reservoirs volume and pumping or chargingperiod as:
Qp =CreservoirsTcharge
(16)
24
The pumping flow is calculated substituting the previous expression for the total volume of reser-voirs (Creservoirs from Eq. 15) and the required charging time (Tcharge = 2h, from RequirementR5.D). The pumping flow rate (Qp) is calculated analogously as the reservoirs capacity (Creservoirs)in the previous section; it is considered the aforementioned range of values for the design head (Hd)and power production (P ). The results are shown in Fig. 7(b), where it is observed that the re-quired pumping flow rate increases with power production and decreases with design head. Theminimum and maximum values of the pumping flow rate are 0.4m3/s and 1.8m3/s for the studiedrange of P and Hd.
Estimating Motor/Generator Characteristics
Typically, the motor/generator is installed in vertical configuration above the turbine runner forvertical shaft turbines, and in horizontal configuration for horizontal shaft Francis turbines. Forfixed-speed pump/turbine, it is required a synchronous motor/generator that works at a fixed speedin generating and pumping modes. It rotates synchronously with the AC grid frequency and it isexcited with DC current (thyristor AC/DC converter required). For variable-speed pump/turbines,it is used a motor/generator with adjustable rotational speed. There are two types of adjustablerotational speed motor/generators: a synchronous motor/generator with variable speed or singly-fed and a doubly-fed asynchronous machine or DFAM [22]. While the DFAM configuration ispreferred for large PHES (> 100MW ), synchronous motor/generator with air/water cooling sys-tem in a closed circuit is chosen for the current design taking as a reference the Pre-FeasibilityStudy of Suduroy in Faroe Islands realised by Norconstult’s [24].
Lots of difficulties have been encountered for the weight and dimension calculations for the genera-tor and the motor, as manufacturing business do not share detailed information about their prod-ucts. An approximate data of m = 1320kg has been found for a 250kW wind-driven generator[31].Due to the lack of more available data, the conservative and approximate calculation of m = 3000kgis made for the motor/generator weight of the Engine Reservoir. No valuable information aboutthe generator and motor dimensions have been found, just what is intuitive from the generatorsimages. From this qualitative information source, the dimensions are set as 2m x 1m x 1m, to bereview when more data is available.
25
8 Elaboration of the design geometry and material
The design geometry selection is made from the literature research and components characterisa-tion of Section 7. As for the material selection, Ocean Grazer experience suggested that the EngineReservoir should be made out of reinforced concrete.
8.1 Characterising the Engine Reservoir walls: thickness and reinforce-ment
An initial guess of the wall thickness and % of steel reinforcement is required to analyse the me-chanical behaviour of the Engine Reservoir. These properties will be optimised afterwards fromthe results analysis in order to satisfy the strength requirements of the structure.
Wall thickness is taken from literature on structures with similar environment. It is taken wallthickness t = 0.9m from the research on the design of a submerged floating tunnel [17], as it issubjected to strong hydrostatic pressures as the Engine Reservoir.
Regarding the walls reinforcement ratio, it is used the approximate data from Tab. 4. It is expectedto obtain higher ratios in further steps of the design, as the Engine Reservoir must withstand sig-nificantly large hydrostatic and earth pressures.
Concrete Building Element Weight of reinforcement (kg/m3)
Bases 90-130Beams 150 - 300Columns 200 - 450Footings 70 - 100Retaining walls 110 - 150Slabs 75 - 135
Table 4: Rule thumb of weight of steel reinforcement per cubic meter of concrete of differentconcrete building elements [29].
8.2 Design I
Following the requirements, the design must contain the elements drawn in Figure 8 and describedbelows:
• Reversible pump/turbines (1) and reversible motor/generators (2) connected through a ver-tical shaft (3): meets the requirements of energy production and storage.
• Penstocks (4): must be inside the Engine Reservoir for better protection to environmentalloads and corrosion and to make possible future design modifications (add penstocks orchange penstock diameter for bigger discharge flow).
• nconnectionRRflow connections with the rigid reservoirs (5) and nconnectionFR
connections tothe flexible bladders (6): the Engine Reservoir shape allows the addition of more connectionsto meet the requirement of future modifications.
• Machine Hall (7): it is defined as the Engine Reservoir area where the power machines areinstalled. It satisfies the requirement of production capacity as it stores the power machines.It also meets the requirement of the lowest high of the pump with respect to the RigidReservoir due to the fact that the Machine Room is buried into the sea bed at a lowest levelthan the RR.
26
• Engine Reservoir Shaft (8): it is defined as the Engine Reservoir area that overcomes thehead difference between the Machine Hall and the Flexible Bladders. It must contain thepenstocks and the nconnections,FB to the flexible bladders (and guarantee the possibility ofadding more flow intakes). Its sectional area must guarantee the machine/operators transportalong the shaft for the requirement of maintenance and replacement operations.
• Exchange Zone (9): it is defined as the most upper zone of the Engine Reservoir. It isadded to the design to meet the requirement of maintenance and replacement operations. Itshould be possible to change the medium of the chamber without altering the medium of theother Engine Reservoir zones. That is to say, it should be possible to change the ExchangeZone medium from air at atmospheric pressure to seawater when machines/operators areintroduced into the Engine Reservoir, and then remove the sea water and recover the airatmosphere to then introduce the machines/operators into the Engine Reservoir Shaft andMachine Hall (and vice-versa). For the design project, it is also considered as an additionalheight of the Shaft, the medium change operations are out of the scope of the project.
zMWL = 0m
zmudline = 35m
zPT = 80m
depth
z (m)
Sea bed
Sea water
A B
C
Air
34
5
2
a
b
c
6
1
7
d
e
Figure 8: Lateral view of the initial shape and structure of the Engine Reservoir. The OceanBattery main systems are itemised using letters: the Flexible Bladders (A), the Engine Reservoir(B) and the Rigid Reservoirs (C). The main components relevant for the requirements satisfactionand for the mechanical analysis are numbered (1-9).
Design variables are defined in Table 5, and shown schematic in the lateral view from Fig. 9. Asobserved in the table, it is chosen that both the Shaft and the Machine Hall have a cylindricalshape. This decision is based on the higher simplicity of the modelling due to the axial symmetry,and it does not interfere with any requirement. Other geometries are considered in Design II.
27
hs
hb
Ds
Db
d
mudline
Figure 9: Schematic lateral view of some initial design variables of the Engine Reservoir.
Design variable Symbol Value UnitEngine Room embedded depth d 50 mConcrete steel reinforcement percentage psr 5 %Shaft shape - cylindrical -Shaft diameter Ds 10 mShaft height hs 45 mShaft wall thickness ts 1 mShaft top covering thickness ttop 2 mMachine Hall shape - cylindrical -Machine Hall diameter Db 15 mMachine Hall height hb 10 mMachine Hall wall thickness tb 1 mMachine Hall base thickness tbase 2 m
Table 5: Initial values of the Engine Reservoir design variables.
28
Figure 10: Top view of the Machine Hall, with the schematic distribution of the main EngineReservoir in colours. The hydraulic and electrical components detailed are: pump/turbine (yellow),motor/generator (orange), penstocks (dark blue), water conveyance system except penstock (lightblue), shaft elevator (green) and additional area for transformers, control system and other otherancillary rooms (gridded region).
8.3 Design II
Due to the difficulty of relating the dimension requirements that derive from the hydro-power com-ponents characterisation (Section 7.2) on a circular area and the foreseen difficulties on designingwall reinforcements for a cylindrical shape, a second design has been constructed, and consideredin most of the models constructed in further sections. The 5 zones described in Design I are man-tained in Design II.
Using the results from Section 7.2 on the main hydro-power machines, a Machine Hall distribu-tion has been developed (Fig. 10). Significant amount of additional space has been left for thenon-characterised componends of the Engine Reservoir (additional area for transformers, controlsystem and other other ancillary rooms), indicated as a gridded region in the figure. Therefore,the design of the Machines Hall is to be optimised when more machine data is available.
The height of the structure is maintained as a design variable depending on the embedded depth.As the Exchange Room must be above the seabed, as well as the flow connections with the FlexibleBladders, the shaft continues above the seabed for additional 2m to account for the flow connectionswith the Flexible Bladder (Requirement R4.A), and the Exchange Room is designed of the samesurface as the shaft (for simplicity purposes) and of 3m of height (enough to fulfill the RequirementsR3.E and R3.F). In conclusion, the squared 4m x 4m that constituted the shaft and the ExchangeRoom reaches a height of 5m above the seabed, and the shaft length in the vertical axis will begiven by the structure embedded depth.
29
9 Literature research: Engine Reservoir interaction loads
According to the requirements in Tab. 1 and the designs description in Section 8.2,8.3, the EngineReservoir interacts with other Ocean Battery components (Rigid Reservoir, Flexible Bladders) andwith its environment (fluid and seabed). This interactions, as all physical interactions, are two-wayinteractions or fully coupled interactions. In order to exemplify the concept, it is considered a fluidflow (component 1) and a flexible structure (component 2) located in a certain way that opposesthe fluid flow. Then, the fluid flow exerts a load on the flexible structure (interaction 1→2), andas a consequence the structure suffers a displacement (result of interaction 1→2). But, at thesame time, the structure displacement on the fluid domain causes a change on the fluid boundaries(interaction 2→1) which results in a change on the velocity field (result of interaction 2→1). It isnow clear that any interaction or action implies a second interaction or reaction. The nature ofthis relations for the Engine Reservoir are detailed in Fig. 11.
It should be noted from the figure scheme that not all the interactions are described, but someinteractions are studied as one-way interactions: from the Fluid to the Engine Reservoir, from theFluid to the Seabed and from the Seabed to the Engine Reservoir. The fully coupled interactionsfound in the real case are simplified as one-way interactions. At this point of the argument, itis worth mentioning that the purpose of the interactions study is not to fully characterise the 3domains (fluid, seabed and structure) but to obtain the boundary loads on the Engine Reservoircaused by the fluid. That is to say, the changes in the Fluid and Seabed are not of significanceif they do not cause, at the same time, a significant change on the Engine Reservoir boundaryconditions. Taking the example of Fluid - Engine Reservoir interaction, the one-way interactionfrom the fluid to the Engine Reservoir will cause small displacements and a stress distributionto the Engine Reservoir. Nevertheless, the effect that this changes will cause to the fluid aresignificantly small (slight change of the velocity field and pressures) and the interaction that thissmall changes in the fluid will cause to the Engine Reservoir are completely neglected. Therefore,one-way interactions are considered because an accurate modelling is only required for the EngineReservoir domain.
According to the previous considerations, the design project and this Section will only describethe one-way interactions shown in Fig. 11. Before the interactions study, the stability conditionswill be introduced for the vertical axis, which considers interactions between all Fluid, Seabed andStructure domains.
9.1 Stability conditions: uplift and driving
The aim of this section is to get an expression of the stability conditions for the phenomena of upliftand driving. Uplift phenomena refers to the structure rising due to buoyancy forces predominance.In contrast, driving phenomena describes the structure sinking due to gravity forces predominance.Therefore, stability in vertical direction is only reached if both uplift and driving conditions aresatisfied. For the stability conditions analysis, it is required the previous calculation of the verticaldeadweight of the structure and of the geotechnical load carrying capacities or resistances.
9.1.1 Structure vertical deadweight
First the vertical deadweight load of the structure must be calculated. The total vertical load of thestructure is WT = mg, where m is the total self-weight of the Engine Reservoir design. It accountsfor the structural concrete weight (Wc),the walls steel reinforcement (Wsr) and the machineryweight (Wm). From the listed weights contribution, the calculation of the total structure self isgiven by:
WT = Wc +Wsr +Wm (17)
It might be necessary to consider an additional deadweight force, the soil filling weight (Ws). Ifapplicable, its exact expression has to be determined from the structure shape and soil properties.
30
Figure 11: Scheme on the interactions between the Engine Reservoir, the other Ocean Batterymain components (Flexible Bladders, Rigid Reservoirs) and the medium domains in direct contactto the Engine Reservoir (fluid and seabed soil).
It should be noted that the soil filling weight already considers the buoyancy effect (as the seabedmudline is bellow the water level or water top). Accordingly, WS is calculated as:
WS = γsVsg − γwVsg = γ′Vsg, (18)
where Vs is the volume of the soil filling (m3), γs is the dry soil density (kg/m3), γw is the seawaterdensity (kg/m3) and γ′ is the so-called effective soil density (kg/m3).
9.1.2 Geotechnical load carrying capacity
The lateral and vertical load capacity of the structure must be calculated for both cohesive andcohesionless soils and compared with the applied loads as stated in the requirements. The load car-rying capacity of vertical and partially embedded piles with closed end (plugged) has been studiedin detail on the field of foundation structures, and its principles are applicable to the cylindricaldesign of the submerged and half-buried Engine Reservoir. In particular, the theory and expres-sions presented along the section are extracted from [1].
The ultimate bearing capacity for axial loading (Qd, N), also referred as soil resistance to drivingor compression (Rs,driving) is determined by:
Rs,driving = Qd = Qf +Qb, (19)
where Qf is the skin or sliding friction resistance (N), Qp is the total end bearing resistance(N).Following this definition, the soil resistance against uplift (due to buoyancy and added loads) isdefined as:
Rs,uplift = Qf . (20)
The skin friction resistance (Qf , N) is calculated through the integration of the unit skin frictioncapacity (f , Pa) along the side surface area (As, m2) as:
Qf =
∫As
f(d)dS. (21)
The integration over the surface is necessary due to depth dependency of the unit skin frictioncapacity (f = f(d)). The different expressions for the unit skin friction capacity are presented
31
bellow for cohesive and cohesionless soils (Eq. 29,32.
The end bearing capacity or resistance (Qb, N) it typically calculated through ultimate Terzaghi’sbearing capacity (qult, Pa) as:
Qb = qultAb, (22)
where Ab is the base or footing area (m2). Terzaghi’s bearing capacity qult is the ultimate bearingcapacity of the structure, that is to say, the smallest bearing load that causes the structure to fail.Then, it might be considered as the maximum bearing capacity, to which a factor of safety will beapplied afterwards (Section 9.1). Terzaghi’s bearing capacity formulation bases on the assumptionthat there is a triangle of pressure just bellow the base of the structure with angle between thebase and the pressure triangle of (45+φ/s). Terzaghi’s bearing capacity considers that the bearingcapacity is developed due to the soil properties of cohesion, friction an density, and it is expressedas [28]:
qult = cNcsc︸ ︷︷ ︸cohesion
+ qNq︸︷︷︸subcharge
+ 0.5BNγγ′sγ︸ ︷︷ ︸
density
, (23)
The cohesion term (cNcs) represents the strength due to cohesion, where c is the soil cohesion(c = 0 for cohesionless soils), Nc is the Terzagui’s cohesion bearing capacity factor (obtained asEq. 24) and sc is the Terzaghi’s cohesion shape factor (obtained from Tab. 6). The subchargeterm (qNq) refers to the bearing capacity strength due to the load subcharge exerted as addedpressure by the soil above the structure base. The variable q is the effective stress at the structurebase or footing (Eq. 28) and Nq is the Terzaghi’s overburden bearing capacity factor (obtained asEq. 25). Finally, the density term (0.5BNγγsγ) represents the strength due to soil density, whereB is the base width or shorter dimension, Nγ is the Terzaghi’s self-weight bearing capacity factor(obtained as Eq. 26), γ′ is the effective soil density and sγ is the Terzaghi’s density shape factor(obtained from Tab. 6).
Terzaghi’s bearing capacity factors Nc, Nq and Nγ are given as a function with respect to theinternal friction angle of the soil (φ, in degrees) [9][13]:
Nc =Nq − 1
tanφ(24)
Nq =e2π(0.75−(φ/360)) tan(φ)
2 cos2 (45 + (φ/2))(25)
Nγ =2(Nq + 1) tan(φ)
1 + 0.4 sin(4φ)(26)
It should be noted that the factor Nc must not be calculated for cohesionless soils, as they havenull soil cohesion per definition (c = 0 for cohesionless soils).
p0 (Pa) is defined as the effective stress for a specific depth h, and it increases linearly with depthas:
p0 = γ′h, (27)
Substituting h for the depth of the structure base or footing (d) in the previous equation, it isobtained the effective stress at the structure base or footing (q, Pa):
q = γ′d, (28)
where, as expected, q increases linearly with the depth of the structure base or footing, also referredas embedded depth.
A. Skin friction and end bearing in cohesive soils
The unit skin friction capacity (f, in Pa) for the case of cohesive soils is calculated through α-Method or Tomlinson Method as [27]:
f = αcu, (29)
32
Terzaghi’s Shape Factors sc sγ
Square footing 1.3 0.8Strip footing 1.0 1.0Round footing 1.3 0.6
Table 6: Terzagui’s shape factors [28]
where α (dimensionless) is the empirical adhesion coefficient that reduces the average undrainedshear strength cu (Pa). The constrained factor α ≤ 1 may be computed as:
α = f(ψ) =
{0.5ψ−0.5 for ψ ≤ 10.5ψ−0.25 for ψ > 1
(30)
where the local variable ψ = cu/p0 is a function of the averaged undrained shear stress (cu) andthe effective overburden pressure at a specific point (p0, defined in Eq. 27).
The undrained shear strength calculation is complex, and diverse models have been developed fordifferent types of cohesive soils from experimental measurements. For a more realistic calculationof cu, the undrained shear strength data is obtained for different cohesive soils (Tab. ??). Inparticular, the table provides the undrained shear strength at mudline (cu,mud) and the rate ofchange of undrained shear strength with depth (mcu). Then, the averaged undrained shear strengthfrom mudline to embedded depth d can be calculated as:
cu = cu,mud +mcud
2(31)
B. Skin friction and end bearing in cohesionless soils
For cohesionless soils, the unit skin friction capacity (Pa) is calculated as:
f = Kpo tan δ, (32)
where K is the coefficient of lateral earth pressure and represents the ratio of horizontal to verticalnormal effective stress, p0 (Pa) is the effective overburden pressure at a specific point (defined inEq. 27) and δ is the friction angle between the soil and the structure wall (degrees). The EngineReservoir has a closed base, therefore it is studied as a plugged case and K = 1 is assumed. If nodetailed measurements are available, φ may be selected from Tab. 2.
C. Skin friction and end bearing limitation in cohesive and cohesionless soils
It is direct to see from Eq. 32, that f increases linearly with the overburden pressure p0, whichincreases linearly with embedded depth (Eq.27). Nevertheless, experimental data show that theskin friction do not increase indefinitely with depth; it reaches its peak value slightly above thetip, and is notably reduced after that. There is no satisfactory theory to analytically predict thenonlinear behaviour of the skin friction with depth, therefore engineers use the approximate theoryof critical depth instead [34]. It states that the unit skin friction capacity f should be limited bya specific value flim. Following this argument, the expression of skin friction capacity (Eq. 32) ismodified as:
f(d) =
{Kγ′d tan δ for d ≤ dlimflim for d > dlim
(33)
where dlim is the limiting or critical depth from where the skin friction capacity is limited. As forthe case of unit skin friction, it is also recommended to limit the unit end bearing value so that itdoes not infinitely increase with depth as the experimental data suggests. There is no valid theoryfor the end bearing capacity behaviour with respect to depth, therefore the critical depths definedfor skin friction capacity limitation are also used for end bearing capacity limitation [34].
33
9.1.3 Stability conditions expressions
When uplift resistance and safety factor are evaluated, buoyancy force must be taken into account.Buoyancy uplift force origins in the water pressure difference around a submerged object. FromArquimedes’ principle, the buoyancy force is equal to the weight of the displaced fluid []:
FB = ρfgV, (34)
where ρf is the fluid density (kg/m3), g is the gravity acceleration (m/s2) and V is the displacedfluid volume.
The whole volume of the structure (both above and bellow the mudline) is considered for thebuoyancy calculation (Eq. 34) as the porous of saturated soil are filled with water (and air in thecase of partially saturated soils). The downward resistance forces that inhibit the buoyancy forcesare the weight of the Engine Reservoir (WT , Eq. 17) and the weight of the soil filling if applicable(WS , Eq. 18). The direction of the shear or skin resistance and the bearing resistance developedby the soil depends on the relation between buoyancy and deadweight forces. For net uplift force(FB > WT + WS), the shear or skin resistance force acts downwards and there is no end-bearingresistance force. Consequently, the soil provides a uplift resistance capacity of Rs,uplift (Eq. 20).Contrarily, when the deadweights overcome the buoyancy and there is a net downwards force(FB < WT + WS), the skin and bearing resistances act upwards. In order to evaluate driving orcompression situation, it has been defined the soil driving resistance capacity Rs,driving. Followingthis argument, the stable equilibrium for vertical loads without considering currents or waves isdescribed through the following inequalities:
FB ≤ (WT +WS) +Qf for FB > WT +WS (35)
FB +Qf +Qp ≥ (WT +WS) for FB < WT +WS (36)
where Eq. 35 refers to the stability against flotation, and Eq. 36 considers the stability againstthe structure sinking into the soil. It is used the skin friction resistance capacity (Qf ) introducedin Section 9.1.2, which is equivalent to the maximum shear resistance that the soil can provide.From the previous stability inequations, the factors of safety for driving and uplift situations(SFdriving, SFuplift correspondingly) are imposed as:
WT +WS +QfFB
= SFuplift ≥ SFuplift (37)
FB +Qf +QpWT +WS
= SF driving ≥ SFdriving (38)
where the values of SFuplift and SFdriving are to be chosen > 1.
It must be noted that the worst case scenario has been considered for the calculation of thebuoyancy force on structures embedded in the seabed. Arquimedes’ principle (Eq. 34) has beenproven to be suitable for structures embedded on saturated sandy soils. Nevertheless, a decrease onbuoyancy force has been observed experimentally for weak-permeable or impervious soils such ascohesive soils due to the interaction between soil and water particle at a micro scale [20]. Therefore,using Arquimedes’ principle for both cohesive and cohesiveless soil is a conservative calculation forthe case of cohesive soils, that adds a safety factor on the anti-floating calculation.
9.2 Seabed → Structure interactions: Earth Pressures
This section focuses on the earth pressure distribution over depth, and its effect on a buried lateralwall. The expressions have been extracted from literature [14]. Depending on the soil-structureinteraction, three types of earth pressures can be considered:
• Active earth pressure: considered when the wall suffers a rotation that causes it to moveaway from the soil in the earth pressure direction. The minimum rotation for active earthpressure is 2mrad (2mm/m, 2mm of lateral movement per meter of the wall height).
34
• Passive earth pressure: considered when the walls moves in the soil direction, opposite tothe earth pressure direction. The minimal rotation for passive earth pressure is 10mrad or10mm/m.
• Earth pressure at rest: is the horizontal pressure acting on the rigid structure. It is consideredwhen the structure does not allow deformations big enough to mobilise active or passive earthpressures.
For simplicity, only earth pressure at rest is applied in the structure. It is computed as:
σr = σzK0 (39)
where σr is the earth pressure at rest (N/m2), σz is the vertical normal total stress or overburdenstress (N/m2) and K0 is the at-rest earth pressure coefficient.The vertical normal stress (σz) is given by:
σz(z) =
∫ z
0
γsdz + γw(z − d) = σ′(z) + u(z) (40)
where γs is the unit weight of the soil (N/m3), specifically the dry soil unit weight for z above thewater level and the submerged or effective soil unit weight (γ′ in Table 3,2) for z below the watertable. γw is the water unit weight (N/m3), z is the soil depth below the seabed surface (m) andd is the depth of the ground water table bellow the soil surface (negative for water level abovethe soil surface). Then, the vertical normal stress can be decomposed into the effective stress (σ′)caused by the soil, and the water pore or neutral pressure (u).The at-rest earth pressure coefficient (K0) is calculated using Terzaghi formula for cohesive soils(Eq. 41) and Jaky expression for cohesionless soils (Eq. 42):
K0,cohesive =ν
1− ν(41)
K0,cohesionless = 1− sinφ (42)
where ν is the Poisson’s ratio and φ is the angle of internal friction of the soil. Jaky’s coefficient(Eq. 42) has been proven to be also valid for normally consolidated clays []. Then, and due tothe high variability of viscosity ν with the soil water content, and the scarcity of viscosity dataavailable for the cohesive soils considered, it has been decided to use Jaky’s expression for bothcohesive and cohesionless soils.Additionally, a surface subcharge is considered to account for the water above the seabed. Thesurface subcharge or vertical surface load causes an increment of the earth pressure (∆σr) uniformlydistributed along the structure wall as:
∆σr = fK0, (43)
where f is the the surface subcharge (N/m2).
9.3 Fluid → Structure interactions: Hydrostatic and hydrodynamicconditions
The fluid exerts an hydrostatic pressure to the structure, perpendicular to its surface. Additionally,other loads have to be taken into account when the hydrodynamic case of currents is considered.Both situations are described in the subsections below.
At the initial phases of the project, when the Engine Reservoir was treated as located on the sea,literature research was done on finding a general methodology for fitting current data into knownvelocity profiles expressions and on the selection of the proper wave governing equations givena certain input data. Nevertheless, this research is if less importance for the lake environmentconditions, where the currents and waves are relatively small, so the mentioned research has beenmoved to Appendix B.
35
9.3.1 Fluid-Structure interaction: Hydrostatic loads
Hydrostatic loads (Fhydrostatic) are constituted by pressure forces that the fluid exerts to the struc-ture in no-flow no-wave situation. It is calculated through the surface integral of the hydrostaticpressure (P ) that acts normal to the body surface:
Fhydrostatic =
∫∫S
Pn dS (44)
where n is the unit vector normal to the surface. Additionally, the value of the hydrostatic pressureonly depends on the depth bellow water level (d) and the fluid density (ρf ), and it is given by:
P = ρfgd (45)
9.3.2 Fluid-Structure interaction: Hydrodynamic or fluid flow loads
The fluid flow exerts a force on the structure surfaces when it passes through it. The hydrodynamicloads can be classified as drag force (parallel to flow direction) and lift force (perpendicular to theflow direction). Furthermore, this forces are the result of two contributors, the pressure force (Fp)and the viscous force (Fv):
Fhydrodynamic = Fp + Fv. (46)
The pressure or pressure-gradient force is generated due to the pressure differences on the bodysurfaces, and can be computed as detailed in the hydrostatic case (Eq. 44). On the other side,viscous force acts on the opposite direction to the fluid flow and it is generated by the actionof the friction force. Both pressure and viscous forces are highly dependant to the flow type. Insome cases one of the pressure of viscous forces is clearly dominant, and the other can be neglected.
The approach of pressure and viscous loads discretisation is used on COMSOL simulations inSection 10.3. Additionally, the analytical model of Morison’s equation is introduced for simulationsverification purposes. The hydrodynamic load model approach uses Morison’s equation to calculatethe horizontal reaction loads on a submerged cylindrical object. The wave contribution is only validwhen the ratio between the waves wavelength and the object diameter is large (> 5). The Morison’sequation for a fixed rigid cylinder is [15]:
Fh(t) = FD(t) + FI(t), (47)
where Fh (N/m) is the horizontal hydrodynamic force vector per unit length acting normal to thecylinder axis, FD (N/m) is the drag force vector per unit length that acts on the cylinder axisand is contained in the plane formed by the member axis and u and FI (N/m) is the inertia forceper unit length that acts normal to the cylinder axis and is contained in the plane formed by thecylinder axis and u.
The drag force per unit length is given by:
FD(t) =1
2CDρDu(t)|u(t)|, (48)
where u(t) (m/s) are the fluid velocity that is horizontal and normal to the cylinder axis, ρ (kg/m3)is the fluid density and D is the cylinder diameter and CD is the drag hydrodynamic coefficient.The horizontal velocity u(t) can also be decomposed as:
u(t) = uw(t) + uc, (49)
where uw(t) is caused by the waves and uc is associated to the current velocity profile and it isassumed constant in time. As for the drag hydrodynamic coefficient, it depends on the shape of theobject that opposes to the flow and the flow velocity (and specifically the flow Reynolds number).There is no exact expression that relates CD with the object shape and flow velocity, but there areplots available that relate CD with the Reynolds number Re. The coefficient CD for the designcase of circular cylindrical object is obtained by substituting Reynolds number (Re, Eq. 50) in Fig.
36
?? for a specific flow velocity and cylinder diameter. The Reynolds number for a circular cylinderis found as [26]:
Re =ρuD
µ, (50)
where rho is the fluid density (kg/m3), u is the flow speed (m/s2), D is the cylinder diameter (m)and µ is the fluid dynamic viscosity ((N·s)/m2 or kg/(m·s)).The inertia force per unit length can be calculated as:
FI(t) = CMρπD2
4u(t), (51)
where CM is the inertia hydrodinamic coefficient and u(t) (m/s2) is the fluid acceleration that ishorizontal and normal to the cylinder axis. For the case of no waves and assumed constant currentvelocity profile, the inertia force is null as the fluid has no horizontal acceleration.
The transverse or vertical force component is characterised by the lift force caused by the largestflow velocities on the top of the structure. When the flow encounters the structure, it must passover the structure or laterally. Then, higher flow velocities on the top of the structure cause lowerpressure, which will induce a vertical lift force perpendicular to the flow velocity (upwards). Thevertical lift force per unit length is given by:
Fv(t) = FL(t) =1
2CLρDu
2(t), (52)
where CL is the lift hydrodynamic coefficient.
9.4 Rigid Reservoirs and Flexible Bladders Connection Loads
The connections loads describe the forces and momentum’s exerted by the Flexible Bladders andthe Rigid Reservoir. For the case of the Rigid Reservoir, it is assumed that dburied is large enoughso that the soil weight and frictions prevent the reservoir from lifting for the critical case of theempty reservoir (when the buoyancy force is maximum). Then, the connection between the EngineReservoir and the Rigid Reservoir just involves fluid flow, and the loads from the Rigid Reservoirto the Engine Reservoir are negligible (FRR−ER ≈ 0N,MRR−ER ≈ 0Nm).
The loads caused by the Flexible Bladder must be analysed in more detail, as it is not buriedinto the seabed. The stakeholder Ocean Grazer is working on the Flexible Bladder subjection onthe seabed, which will be done through a system of cables supported by embedded anchors or the(buried) Rigid Reservoir. It is worth highlighting that the Flexible Bladder net vertical force isrelatively small, as the buoyancy force (Fb) is expected to be very similar to the gravity force (Fg).As the flexible hermetic reservoir adapts its volume to the volume of contained water, it shouldnot contain air at any moment of the Ocean Battery working cycle. Then, the buoyancy net force(Fb−Fg) is only caused by the small difference of density between the exterior lake water and theinterior fluid, and the relatively thin exterior layer of the flexible bladder. For the case of the fluidwithin the flexible bladder, stakeholder Ocean Grazer have chosen fresh water. As the fluid insidethe Flexible Bladder is the same as the exterior lake fresh water, there is no net buoyancy force.Consequently, the bladder net buoyancy force is caused only by the bladder material as:
Fnet = Vbladderg(ρw − ρbladder), (53)
where Vbladder is the thickness volume of the bladder membrane, g is the gravity accelerationand ρw and ρbladder are the densities of the water and the bladder material correspondingly. Forthe Flexible Bladder material, previous Ocean Grazer projects worked with Ethylene PropyleneDiene Monomer rubber (EPDM) [21] of density ρ = 20kg/m3[3]. No information have been foundregarding the relation between EPDM material and the Flexible Bladder capacity. Moreover,commercially available pillow tanks have a maximum capacity of 2000m3 approximately, which issmaller than the minimum capacity required of 3000m3 (Fig. 7(a)). Nevertheless, for the sake ofgetting an approximate idea of the order of magnitude of the net buoyancy force on the flexible
37
bladder, it is assumed that the Flexible Bladder resembles a pillow tank of capacity 3000 ∼ 6000m3
(Fig. 7 and P = 1MW ). Linyi Solpac Co. claims to be able to produce a pillow tank of 100m3 ofthickness d = 1.2mm[8]. Roughly scaling the thickness to d = 36mm for a pillow tank of capacity3000m3, and choosing a square base pillow tank of dimensions [L x W x H] = 45m x 45m x 1.5m(capacity of 3037m3), the bladder volume can be approximately calculated as:
Vbladder = (L+ 2d)(W + 2d)(H + 2d)− LWD. (54)
Substituting the pillow tank parameters into Eq. 54 and Eq. 53, it is obtained Vbladder = 77.88m3
and Fnet = 7.49 · 105N ≈ 750kN . Although it might seem a significantly large force, the RigidReservoir withstands a buoyancy force caused by O(103)m3 of air (3000m3 in the case of P=1MWand dburied = 50m), while the Flexible Bladder net buoyancy is only caused by 77.88m3 of bladdermaterial, which has a density relatively close to air density. Therefore, it is considered that theRigid Reservoir has very large axis bearing capacity and can make the function of retaining theFlexible Bladder through a net of ropes (or similar) that are subjected by the Rigid Reservoir. Asa conclusion, the connection loads or interaction between the Flexible Bladders and the EngineReservoir is also neglected.
38
10 Model setup
10.1 Model I. Stability conditions on structure uplift and driving andsoil load carrying capacity (Matlab)
Model I analyses the structure stability of Design I regarding the phenomena of uplift and drivingand with respect to the structure embedded depth. Therefore, the uplift and driving safety factorsare compared with the existing vertical forces using the inequations from Section 9.1. To do so, thegeotechnical load carrying capacity, the sliding friction resistance and the end bearing resistanceare calculated as in Section 9.1.2. The structure geometry is taken from Tab. 5, and its locationwith respect to the mudline is taken as a variable in order to study at a embedded depth influenceover uplift and driving phenomena (dburied ∈ (0, 50)m). Additionally, a parametric study on thesatisfaction of the safety factors is performed for 7 types of soil (Tab. 2, 3).
Design I uses reinforced cement concrete for all the structure elements (Section 8.2). Model Iweight calculations use concrete density ρc = 2.400kg/m3 and approximated reinforcement ratiosper structural element from Tab. 4. For the sake of simplicity and due to uncertainty on DesignI, it is only considered the building elements of Bases, Slabs and Retaining Walls.
The results from Model I have been analysed previous to the construction of the next models. Anadequate embedded depth dburied is chosen from the study of Model I results, and it is taken asinput data by the next models. Additionally, all the next models are done with Design II, whichhas more optimised dimensions than Design I.
10.2 Model II. Fluid → Porous Seabed Interaction (Comsol)
The model studies the effect of soil saturation on the soil. Soil saturation is the phenomena ofa fluid filling the soil porous or space between soil particles, and it appears when the soil is lo-cated below the water level or water table. Then, the fluid fills the soil porous driven by gravityforces (free flow) or due to applied load or pressure (forced flow). A completely saturated soil,or a soil of 100% saturation, describes a porous soil where all the porous are filled with water.In the studied case of the Engine Reservoir environment, the porous soil is located on the lakebottom. The seabed surface is in direct contact with the fluid, and the fluid exerts a large andevenly distributed hydrostatic pressure on the seabed. Therefore, it is expected that the soil iscompletely saturated. Nevertheless, the different layers of soil might produce alterations into thefluid flow and pore pressure. The extreme case is when the soil contains impermeable layers; thefluid cannot penetrate them, and therefore the pore pressure drops to zero.
Model II works with the limit case of completely saturated soil. It considers only one layer of soil(constant characteristics with depth) and it is characterised as porous and permeable. Additionally,the approximation of completely horizontal soil is used, which makes possible to use a 2D spacewhere the second coordinate y accounts for depth. The setting for the model geometry, material,physics and multi-physics are described below:
• Geometry: 2d space with fluid domain (D2) and porous soil domain (D1). The domainsdimensions account for the water level of 35m and the maximum buried depth of 50m (Fig.12).
• Materials: water (fluid) for domain D2 and seabed soil (porous media) for domain D1. Theseabed soil materials considered are Kaolin Clay and Silica Sand, the properties relevant forthe studies are detailed in Table 7.
• Physics: Free and Porous Media Flow with Free and Porous Media Flow interface (D1 andD2) and Solid Mechanics (D1). The physics boundary conditions described in Tab. 8 usingthe boundary numbering detailed in Fig. 12.
• Multiphysics: Free and Porous Media Flow and Solid Mechanics interfaces are coupled.The boundaries where multiphysics is applied are detailed in Tab. 8 using the boundarynumbering detailed in Fig. 12.
39
Figure 12: Geometry and free and porous media flow boundary conditions of model I. Boundarynumbering in blue.
Parameter Name Symbol Kaolin Clay Silica Sand Unit
Effective soil weight γ′ 5993 6500 [N/m3]Young Modulus E 10 - 200 10 - 50 [MPa]Poisson’s ratio ν 0.2 - 0.4 0.15 - 0.35 -Porosity ns 33-60 20-35 %Permeability ks 10−9 - 10−6 10−5 - 10−3 [cm/s]
Table 7: Soil parameters used on Model II [4]
It should be noted that COMSOL Multiphysics includes several interfaces to study the fluid flowwithin a porous media, each one based in different equations and assumptions. Within the physicsof Free and Porous Media Flow, the following interfaces can be used: Darcy’s Law, Fracture Flow,Richard’s Equation, Brinkman Equations, Free and Porous Media Flow, Phase Transport in PorousMedia and Multiphase Flow in Porous Media [10]. Fig. 14 shows the adequate governing equations(used by the before mentioned interfaces) of each soil region. From the figure, it is concluded thatthe adequate governing equation are Navier-Stokes equations for the fluid domain and BrinkmanEquations for the porous soil domain. Then, the interface Free and Porous Media Flow is suitablefor the model, as it solves Navier-Stokes equations in the fluid and Brinkman equations in the soil.
10.3 Model III. Fluid → Structure Interaction (Comsol)
Model III aims to study the loads and moments on the structure caused by the fluid hydrostaticloads (pressure) and the hydrodynamic loads derided from the current. Furthermore, the impor-
Boundary Free and Porous Media Flow (D1,D2) Solid Mechanics (D1)
1 Wall, slip / Symmetry Roller / Symmetry2 Wall, no-slip Fixed Constant3 Wall, slip / Symmetry -4 Multiphysics, fully coupled Multiphysics, fully coupled5 Open boundary, No Normal Stress -6 Wall, slip / Symmetry Roller / Symmetry7 Wall, slip / Symmetry -
Table 8: Boundary conditions of the physics in Model II, using the boundary nomenclature detailedin Fig. 12
40
Figure 13: Schematic view of the cross-section of a river bank. Specified the governing equationsof the fluid flow for each soil region [].
Figure 14: Geometry and boundary conditions of Model III.
tance of current modelling for the purpose of the Engine Reservoir analysis will evaluated throughthe comparison between hydrostatic pressure derived forces and current derived forces.
As stated in Section 9.3, the interaction between fluid and structure is not fully coupled but sim-plified as one-way interaction (from the fluid to the structure). Then, only the effects of the fluidto the structure are considered (such as hydrostatic pressure, viscous forces) and the effects of thestructure to the fluid (such as fluid field change due to structure displacement) are neglected. As aresult, it is enough to model only the fluid domain in order to evaluate the effect of the fluid to thestructure. Accordingly, the model consists on one 3d fluid domain, and the structure is added inthe form of domain difference. The Engine Reservoir above the mudline consists on the upper partof the shaft, with 5m of height and a square planar area of 4m x 4m; the fluid domain differenceis relatively small compared with the size of the fluid domain. It is considered the limit case whenthe current is perpendicular to one of the structure faces, as it is the case when the interactionloads are expected to be more significant. Then, it can be used the simplification of only modellinghalf of the domain on y-axis due to symmetry on the xz-plane (using coordinate system from Fig.14).
The model geometry is a single fluid domain of dimensions 200m x 15m x 35 m, with a differenceof dimensions 5.4m x 2.7m x 5m and centred on (100,0,0)m that represents (half) of the structureabove the mudline. Fig. 14 shows the orthographic projection of the geometry. As for the physicsutilised, the fluid domain is modelled as Turbulent Flow with k and ε as turbulent parameters.The boundary conditions are no sliding wall for domain bottom (fluid boundary with the seabed),sliding wall for the upper (water surface) and lateral boundaries, and inlet and output for opposingboundaries perpendicular to x-axis (Fig. 14). The inlet velocity field is chosen perpendicular tothe boundary surface, and it is used a normal state velocity profile. In order to get the velocityprofile, data from Appendix A is taken for the currents normal state, and it is fitted in Section ??as predominantly tidal current profile.
Some comments should be done on the subject of fluid forces calculation method. From the solutionto the microscopic momentum balance or Navier-Stokes equation, it is found the body surface loads
41
caused by the fluid flow (F ) []:
F =
∫∫s
(n ·Π)|surfacedS, (55)
where Π the total stress 2nd order tensor, n is the unit normal vector to infinitessimal surfacedS and S is the macroscopic body surface that interacts with the fluid. In accordance with thehydrostatic loads description in Section 9.3.2, the total stress tensor have the contribution of aviscous term and a pressure term:
Π = τ − pI, (56)
where τ is the viscous stress tensor, p is the fluid pressure and I is the identity tensor. Finally, forNewtonian fluids (such as water) the viscous stress tensor depends on fluid velocity as:
τ = µ(∇v + (∇v)T ) (57)
In Comsol implementation, total force is calculated through the surface integration of T , the totalstress vector on a surface at a particular location with unit normal vector:
T = (n ·Π)|surface =
spf.T stressxspf.T stressyspf.T stressz
(58)
Analogously, the viscous force is calculated through the surface integration of K, the viscous stressvector on a surface at a particular location with unit normal vector n:
K = (n · τ)|surface =
spf.K stressxspf.K stressyspf.K stressz
(59)
10.4 Model IV. Porous Seabed → Structure Interactions (Matlab)
Model IV studies the stress loads caused by the porous soil to the structure surfaces. Specifically,it is calculated the Lateral Earth Pressure at-rest on the structure lateral surfaces and the porewater pressure over soil depth. The model is governed by the soil equations presented on Section9.2. For the sake of briefly, only two types of soil have been tested by the model: Kaolin Clayand Silica Sand (Tab. 2,3). This two soils have been selected because they present significantlydifferent properties; the results comparison will provide information about which are the significantproperties on soil pressures calculation and the nature of its influence. By assuming completelysaturated soil, the vertical normal stress is calculated as in Eq. 40. For the water pore pressurecalculation, it has been considered the soil top level at 35m below the water level (zmudline = 35m).For accounting for the extra pressure due to the 35m of water above the soil, it is equivalent tocompute pore pressure taking d = −35m in Eq. 40 than taking d = 0m and adding an uniformlydistributed load with a surface subcharge of f = γwgzmudline in Eq. 43.
10.5 Model V: Integrated Model (Comsol)
Model V is called Integrated Model because it accounts for the interactions between Fluid, Soiland Structure, studied in previous models. Each medium or domain is driven by a set of governingequations, that can be implemented in COMSOL through Physics and Physics Interfaces. Theinteractions between mediums are introduced by the determination of boundary conditions and byMultiphysics interface. The governing equations of each domain or sub-model are described in thefollowing subsections 10.5.1, 10.5.2 and 10.5.3.
The three numerical sub-models described below are to be integrated in COMSOL using ArbitraryLagrangian-Eulerian (ALE) mesh (moving mesh), boundary conditions described in Section 10.5.5,seabed parameters from Table 9 and the following Physics:
• (Fluid domain) ’Laminar Flow’ Physics with incompressible fluid for Navier-Stokes sub-model
42
Figure 15: Geometry and boundary conditions of Model V
• (Seabed domain) ’Porous Media and Subsurface Flow’ Physics with ’Darcy’s Law’ or ’Freeand Porous Media Flow’ interfaces.
• (Structure domain) ’Solid Mechanics’ Physics
• (Interaction between Physics) ’Poro-elasticity’ Multiphysics
Unfortunately, the COMSOL Multiphysics licence provided by University of Groningen do notinclude the licence for ’Subsurface Flow Module’, required for the seabed Physics and the ’Poro-elasticity’ Multiphysics. Consequently, it has been taken an alternative approach for seabed mod-elling, which consists on introducing the seabed sub-model governing equations (Section 10.5.2) inthe form of Partial Differential Equations (Section 10.5.4) using ’Coefficient Form PDE’ Physics.Nevertheless, no results are presented for Model V because the Stationary Study did not converge.At this point, Model V has been simplified as Model VI in order to obtain study convergence andvaluable results.
Variables Symbol Value UnitPorosity ns or ε 0.4 -Permeability ks 10−3 m/sShear modulus G 5 · 108 N/m2
Poisson’s ratio µs 0.33 -Saturation S 0.98 -True modulus of water elasticity Kw 2 · 109 N/m2
Table 9: Model VI porous seabed parameters
10.5.1 Governing equations of the fluid sub-model
Fluid velocity field and pressure distribution is described by Navier-Stokes equations (NS). Insome particular where turbulence phenomena plays an important role, might be adequate to usethe time-averaged Reynolds-Averaged-Navier-Stokes equations (RANS). Through the impositionof momentum balance in terms of stress, it is obtaned the generalised equations of the fluid flowin terms of velocity gradients and fluid pressure:
ρ∂~u
∂t−∇ · [η(∇~u+ (∇~u)T )] + ρ(~u · ∇)~u+∇pf = ~F , (60)
∇ · ~u = 0, (61)
where ρ is the fluid density, η is the fluid dynamic viscosity, ~u is the fluid velocity field, pf is the
pressure and ~F is a volumetric force, the gravity in the case studied.
43
10.5.2 Governing equations of the seabed sub-model
Seabed sub-model bases on Biot’s poro-elastic theory [] , which treats the seabed as a compress-ible porous medium with a flow of a compressible pore fluid. The governing equation for massconservation is given by:
∇ · (K∇ps)− γwnsβs∂ps∂t
= γw∂εs∂t
(62)
where ∇ is the divergence operator (∇ = ∂∂x + ∂
∂y + ∂∂z ), ps is the pore pressure (Pa), γw is the unit
weight of the pore water, ns is the soil porosity, βs is the compressibility of the seabed pore waterpressure and εs is the volumetric strain. Assuming the seabed to be homogeneous and isotropic,that is to say, with the same permeability ks in all directions, Eq. 62 can be rewritten as:
∇2ps −γwnsβsks
∂ps∂t
=γwks
∂εs∂t
, (63)
where ∆ is the Laplace operator (∆ = ∂2
∂x2 + ∂2
∂y2 + ∂2
∂z2 ). The compressibility of the pore fluid canbe denoted as:
β =1
Kw+
1− SPw
, (64)
where Kw is the true modulus of water elasticity (N/m2), S is the soil degree of saturation andPw0 is the absolute static water pressure. The volumetric strain is defined as:
εs = ∇ · ~us =∂us∂x
+∂vs∂y
+∂ws∂z
, (65)
where ~us = (us, vs, ws) is the soil displacement vector and its decomposition in xyz-direction.
The relationship of stress-strain (between pore pressure and soil displacement) for the force equi-librium is given by:
G∇2~us +G
1− 2µs∇εs = ∇ps, (66)
where G is the soil shear modulus and µs is the soil Poisson’s ratio. Eq. 66 can be decomposed as:
G∂2us∂x2
+G
1− 2µs
∂εs∂x
=∂ps∂x
(67)
G∂2vs∂y2
+G
1− 2µs
∂εs∂y
=∂ps∂y
(68)
G∂2ws∂z2
+G
1− 2µs
∂εs∂z
=∂ps∂z
(69)
and the shear modulus G is related to the soil Young’s modulus (E) and the Poisson’s ratio (µs)as:
G =E
2(1 + µs)(70)
The equilibrium equations in the three coordinate directions that impose force equilibrium aregiven by []:
∂σ′x∂x
+∂τxy∂y
+∂τxz∂z
=∂ps∂x
(71)
∂τyx∂x
+∂σ′y∂y
+∂τyz∂z
=∂ps∂y
(72)
∂τzx∂x
+∂τzy
∂y+∂σ′z∂z
=∂ps∂z
(73)
44
where σ′x, σ′y and σ′z are the effective normal stresses in xyz-direction, τxy, τxz, τyx, τyz, τzx and
τzy are the effective shear stresses and ps is the pore pressure. Then, the expressions of effectivenormal and shear stresses can be obtained by substituting Eq. 71-73) in Eq. 67-69:
σ′x = 2G[∂us∂x
+µs
1− 2µsε]
(74)
σ′y = 2G[∂vs∂y
+µs
1− 2µsε]
(75)
σ′z = 2G[∂ws∂z
+µs
1− 2µsε]
(76)
τxy = τyx = G[∂us∂y
+∂vs∂x
](77)
τxz = τzx = G[∂us∂z
+∂ws∂x
](78)
τyz = τzy = G[∂ys∂z
+∂ws∂y
](79)
It is worth mentioning that the governing equations regarding the force balance use the Terzaghi’sprinciple of effective stress. It states that the soil total stress (σij) can be decomposed into the soileffective stress (σ′ij) and the soil pore pressure (ps) as:
σij = σ′ij + δijps, (80)
where δij refers to the Kronecker delta.
10.5.3 Governing equations of the structure sub-model
The structure sub-model bases on small-displacements assumption. It is used the following rela-tionship between strain and displacement components:
εx =∂u
∂x, εxy = εyz =
1
2
(∂u∂y
+∂v
∂x
), (81)
εy =∂v
∂y, εyz = εzy =
1
2
(∂v∂z
+∂w
∂y
), (82)
εz =∂w
∂z, εxz = εzx =
1
2
(∂u∂z
+∂w
∂x
), (83)
where the displacement field is ~u = (u, v, w) in the system coordinates (x, y, z) and the straintensor ε is given by:
ε =
εx εxy εxzεyx εy εyzεzx εzy εz
. (84)
Analogously, the structure stress tensor is defined as:
σ =
σx τxy τxzτyx σy τyzτzx τzy σz
. (85)
Using the strain and stress tensors described above, the structure behaviour is modelled using theassumption of linear elastic behaviour between stress and strain:
σ = Dmε, (86)
where Dm is the elasticity matrix, and the weak formulation of the equilibrium conditions in termsof stress:
−∇ · σ = ~F , (87)
where ~F accounts for volume or body forces.
45
10.5.4 PDE Coefficients for seabed sub-model
Governing equations of the seabed sub-model (Section ??) are implemented in Comsol Multiphysicsthrough Physics module ’Coefficient Form PDE’. With pore pressure (ps) as dependent variable,the general expression of the PDE (Partial Differential Equation) is:
ea∂2ps∂t2
+ da∂ps∂t
+∇ · (−c∇ps − αps + γ) + β∇ps + aps = f, (88)
where c is the diffusion coefficient (m/s), a is the absorption coefficient (1/m·s), f is the sourceterm (kg/m2·s3), ea is the mass coefficient (s/m), da is the damping coefficient (1/m), α is theconservative flux convection coefficient (1/s), β is the convection coefficient (1/s) and γ is theconservative flux source (W/m3). The coefficient values are chosen to so that the general PDE(Eq. 88) represents the governing equation for mass conservation from Biot’s poro-elastic theoryand for isotropic seabed (Eq. 63). Then, all coefficients are chosen null except for:
c = −ks (89)
da = −γwnsβs (90)
f = γw∂εs∂t
(91)
10.5.5 Boundary Conditions
Define boundary conditions, relate with sub-models
• Structure - Seabed: no-slip boundaryIt is assumed that the structure has the same displacement and velocity as the seabed, thereis no relative displacement between seabed and structure (Eq. 92). Additionally, as thestructure is considered rigid and inelastic, there is a null normal gradient of pore waterpressure (Eq. 93). The total stress equilibrium at the boundary is given by Eq. 94.
us = ust (92)
∂ps∂n
= 0 (93)
σst = σ′s, τst = τs (94)
where subscript ’s’ accounts for soil domain, and ’st’ for structure domain.
• Structure - Fluid Boundary: no-slip boundaryThe structure normal stress is determined by the fluid (Eq. 95), while the moment shearstress is neglected (Eq. 96).
σst = −pf (95)
τst = 0, (96)
where Pf is the normal stress that the fluid exerts normally to the structure.
• Fluid - Seabed (seabed surface): no-slip boundaryIt is assumed that the soil has null vertical effective stress and null shear stress (Eq. 97) andthat the pore water pressure of the seabed is given by the fluid pressure on the boundary(Eq. 98).
σ′s = 0, τs = 0 (97)
ps = pf (98)
• Seabed underlying bottom: it is assumed that the seabed boundaries with an impenetrablematerial; there are no displacements, no pore water pressure gradient and no normal verticalflow.
us = vs = ws = 0 (99)
∂ps∂n
= 0 (100)
46
• Seabed side boundaries: it is assumed zero displacement for side boundaries sufficiently farfrom the embedded structure.
• Fluid inlet: (piston) wave-maker generates the waves. Constant current velocity profile overtime.
• Fluid outlet: sponge layer to reduce the wave reflection.
• Fluid surface: free boundaryFree boundary implies zero pressure applied (except from a reference pressure of pref =1atm).
10.6 Model VI - Integrated model II (Comsol)
The attempt to simultaneously model and simulate all the mediums (fluid, seabed and structure)and its interactions through Model V has failed. Therefore, Model VI has been developed as asimplification of Model V. It models only the structure domain, but it applies as boundary loadsthe known one-way interactions between mediums studied in previous models.
Model VI works in 3D space and models the structure geometry using Design II parameters andembedded depth dburied = 30m below seabed surface. As for the materials, the initial approachwas to directly model the reinforced concrete, combining concrete and steel domains and takinginto account the interactions between them through Solid Mechanics boundary conditions. Never-theless, this procedure has been found not adequate for COMSOL Multiphysics due to convergenceerrors and high uncertainty of the geometry of the reinforcement. Consequently, the structure hasbeen modelled using pure concrete material. The result analysis will provide information whetherreinforcement is necessary for the given structure design.
The Physics used is Solid Mechanics, and no Multiphysics interfaces are required. The boundaryconditions on the structure-fluid boundary (z > 0m) are extracted from the results of Model III(Sections 11.3, whose analysis conclude that he fluid loads to the structure are mainly due tohydrostatic pressure, and that the effects of fluid flow can be neglected. Accordingly, the boundarycondition for structure surfaces for z > 0m, being z = 0m the height of the seabed surface, is givenin terms of pressure as:
P (z) = γwg(dseabed − z), (101)
where dseabed = 35m is the depth of the seabed surface with respect to the water level. Theboundary conditions on the structure-seabed boundary (z < 0m) are taken from Model IV results.Accordingly, the so-called lateral earth pressures at-rest are applied on the structures surfacesbelow the seabed surface (z < 0m). Additionally, for the horizontal structure surface buriedon the seabed (ceiling of the Machines Hall) no lateral earth pressure has been applied but thecombination of pore pressure and filling soil weight. It is worth mentioning that the soil weightis calculated using the effective or submerged soil weight, referred as γ′ along the report. Allboundary conditions are represented graphically in Fig. 16.
47
(a) Vertical earth pressures for z < 0m and verticalhydrostatic pressures for z > 0m.
(b) Lateral earth pressures at-rest for z < 0m and hor-izontal hydrostatic pressures for z > 0m.
Figure 16: Surface boundary loads as pressures applied on the structure external surface due tothe one-way interactions of the fluid and seabed to the structure.
10.7 Model VII - Reinforced structure (Cypecad)
Finally, Model VII is developed with the goal of providing a complete mechanical study thattakes into account the surface loads exerted by the environment, the structure self-weight andhydropower components loads and the steel reinforcement contribution. It uses dburied = 30mand dseabed = 35m for loads computation and Silica Sand (Table 2) for soil thrust calculation.Nevertheless, the model allows to change these design and environment parameters easily.
The Engine Reservoir is structured in 5 levels or floors, as shown in Figure 18. The height,description and structural loads of the floors are detailed in Table 10. The table differentiates twotypes of structural loads:
• Dead loads: also known as permanent or static loads, they are the relatively constant loadsover time.
• Life loads: also known as imposed loads, they refer to dynamic loads or temporary loads.They are the variable forces over the normal operation cycle of the structure.
As observable in the table, Cypecad works with superficial and evenly distributed loads in t/m2,where t accounts for tones. The superficial dead loads for the Engine Reservoir ceiling P5 (30t/m2)and for the Machines Hall ceiling P3 (46t/m2) are directly obtained using the expressions for hy-drostatic pressure and the (effective) soil weight above P3. The life load applied on the ExchangeRoom Floor accounts for replacement operations of hydro-power machines. Assuming a movingmechanism that works as an elevator along the Engine Reservoir Shaft, the elevator temporary loadhas been chosen as the weight of 5 persons and 1 hydropower machine. Using the approximate val-ues of 80kg/person and 600kg/machine (extracted from reversible pump-turbine weight calculationin Section 7.2, the total temporary weight is 1000kg = 1t. In order to account for the self-weightof the elevator structure, an additional 1t has been added to the calculation. Then, the surfacelife load is 2t evenly distributed on a 4m x 4m surface, which results in a surface load of 0.13t/m2.Finally, P2 dead load of 1t/m2 is obtained from the weight of the hydro-power machinery andwater conveyance system. Using the approximation that the weight is evenly distributed alongthe floor, and that all the penstock weight (and filling water) is supported by the floor. Assuming
48
Concrete structures
beamspilarswallsfoundationssolid slabs
Figure 17: Cypecad 3D representation of the Engine Reservoir model.
FloorName
Height[m]
Level[m]
Length x Depth[m x m]
Life Loads(L) [t/m2]
Dead Loads(D) [t/m2]
P5 3 5 [4 x 4] 0 30P4 28 2 [4 x 4] 0.13 0P3 3 -26 [12 x 8] + [4 x 4] 0 46*P2 1 -29 [12 x 8] + [4 x 4] 0 1*P1 - -30 [12 x 8] + [4 x 4] 0 0
Table 10: CYPECAD floors description. *Surface loads only applied on [12 x 8] floor region.
maximum penstock diameter (Dp,max ≈ 1m from Section 7.2), penstock completely full of fluid, 4reversible pump-turbines
49
Shaft
Exchange Room
Machines Hall
Foundations
Height (m)
5
20
-26
-29-30
P5
P4
P3
P2 P1
Figure 18: Visualisation of Cypecad floors. Height defined with the origin in the seabed surface.
50
(a) (b)
Figure 19: Parametric study on ER embedded depth of static ER deadweights, buoyancy and soilcapacities. Soil considered: Silica Sand, cohesionless.
11 Simulation Results
11.1 Results Model I
The results for (cohesionless) Silica Sand are presented in detailed in Fig. 19, 20. It is easy to ob-serve the bearing capacity Qp is significantly bigger than the rest of vertical forces and capacities,therefore the safecty factor against buoyancy is satisfied without problems (SF driving > SFdrivingfor all embedded depths d ∈ (0, 50)m). Converselly, the safety factor against driving or sinking isonly satisfied for embedded depths bigger than a critical value, in this case dcritical ≈ 20m (whereSFuplift > SFuplift).
From the results presented for Silica Sand, it is decided to compare the critical requirement (sat-isfaction of SFuplift) for multiple cohesive and cohesionless soils. The results are compared inFig. 21. It is observed that Design I must be embedded at least 15m to satisfy the safety factoragainst uplift. Additionally, cohesionless soils show better resistance against uplift, that is to say,the minimal embedded depths that satisfies uplift safety factor is smaller for cohesionless soils.Additionally, it is also observed that Nkossa Clay and Kaolin Clay perform the worst in the upliftsafety factor satisfaction and they are also the less dense cohesionless soils.
51
Figure 20: Parametric study on ER embedded depth of stability safety factors against uplift anddriving. Soil considered: Silica Sand, cohesionless.
Figure 21: Parametric study on ER embedded depth of stability safety factor against uplift(SFuplift) for multiple cohesive (continuous curves) and cohesionless (dotted curves) soils.
52
11.2 Results Model II
Model II results on Figure 22 show the expected behaviour of the effective vertical stresses andpore pressure. The pore pressure increases linearly with depth, as expected from the expressions onSection 9.2. Additionally, Figure 22 show the linear increase of the effective earth stresses from themudline, also as expected from Section 9.2 expressions. It is worth mentioning that the relativelysmall pressures of silica sand are caused by its relative smaller density with respect to Kaolin Clay.It is also observed a large difference between the soil effective stresses (continuous lines) and thelateral earth stresses at-rest (dashed lines).
Additionally, soil displacement along y-coordinate is studied for the soils ranges of Young Modulus(E). It is clear to see from Fig. 23 that the soil is suffering a compression, and furthermore this soildisplacement is inversely proportional to Young Modulus. The displacement order of magnitude ifof O(-1).
11.3 Results Model III
11.3.1 Fluid-Structure Interaction: Hydrostatic loads
11.3.2 Fluid-Structure Interaction: Hydrodynamic loads
The hydrodynamic pressure and viscous forces introduced in Section 9.3.2 are calculated in Comsolas detailed in Section 10.3. The values of the total load and its contributions (pressure and viscousforces) are calculated for the whole body (integration over all the surface) and for certain structuresurfaces in order to get a better understanding of the results.The total hydrodynamic force (F ) for all structure surfaces and its contributions, the pressureforce (Fp) and the viscous force (Fv), are:
F =
289.300
−8.5632e6
N, Fp =
288.270
−8.5632e6
N, Fv =
1.02690
0.13572
N (102)
where the y-component of the loads is null due to the structure symmetry in y axis and the con-sideration of perfectly perpendicular flow in x-direction.
The surface loads results for each face are presented individually in Table 11. Small calculationshave been made to extrapolate the half-space Comsol simulation to the results for each face, asthe Comsol result only provided with the loads recieved in half of Face A,B and C and all FaceD. Therefore, for Face A,B and C and for all the loads (Fp, Fv and F ), the x,z-components havebeen multiplied per 2, and the y-components have been set to 0 per symmetry. This calculationscan only be made for fluid flow in x-direction and perpendicular to Face A, as in the case of thesimulation.
The Comsol simulation gives as a result the fluid pressure in the hydrodynamic case in all fluiddomain (p). This result is compared with the theoretical hydrostatic pressure (p∗), computedanalytically as in Eq. 45. Both pressures are compared through their difference (p − p∗), as inFig. 27 for a certain vertical line. Moreover, the integrated values of the (analytical) hydrostaticpressure in the structure surfaces are compared with the integrated values of the pressure differencein the same structure surfaces. The obtained integrated values differ by 3 orders of magnitude:
Half Face A: p∗ = 4.2964e6Pa, p− p∗ = −1.3734e3Pa (103)
Half Face C: p∗ = 4.2832e6Pa, p− p∗ = −1.5607e3Pa (104)
Face D: p∗ = 8.5928e6Pa, p− p∗ = −3.0992e3Pa (105)
(106)
53
Figure 22: Effective vertical stress and lateral earth stress at-rest for Kaolin Clay (black) and SilicaSand (green). Pore water pressure (blue) for both types of soil.
(a) Seabed soil: Kaolin Clay (b) Seabed soil: Silica Sand
Figure 23: Soil displacement along soil depth for different values of Young Modulus for soil type(a) Kaolin Clay and (b) Silica Sand.
54
Pressure Force (N) Viscous Force (N) Total Force (N)
Face A
8.590e60
−2.785e−11
0.2950
−0.096
8.590e60
−0.096
Face B
−8.590e60
1.330e−8
−1.388e−40
0.121
−8.590e−60
0.121
Face C
4.204e−150
−8.563e6
0.5140
0.107
0.5140
−8.563e6
Face D
4.204e− 15−8.590e6−1.132e−9
0.1090.1050.002
0.109−8.590e6
0.002
Table 11: Caption
(a) Velocity field yz-plane,x component (m/s)
(b) Velocity fiels yz-plane,y component (m/s)
(c) Velocity field yz-plane,z component (m/s)
Figure 24: Plane representation of the fluid velocity field. Decomposition of the fluid velocity fieldin x,y,z-components in yz-plane in x=100m (structure centre in x-direction).
55
(a) Velocity magnitude xy-plane, z=3.33m, zoomed in
(b) Velocity magnitude xz-plane, y=0m
Figure 25: Plane representation of the fluid velocity on significant xy and xz-planes. Velocitymagnitude (m/s) surface and velocity field streamlines.
(a) Total stress, x component (N/m2) (b) Total stress, y component (N/m2)
(c) Total stress, z component (N/m2)
Figure 26: Total stress distribution on the structure surface on interaction with the fluid. Decom-position of the total stress in x,y,z-direction.
56
Figure 27: Pressure difference between the simulated hydrodynamic pressure (p) and the hydro-static pressure (p*) calculated analytically (Eq. 45). The hydrodynamic pressure profile (p(z)) istaken from the simulation for x = 97.3m and y = 0.01 and z ∈ (0, 35), which corresponds to avertical line along the fluid domain and that is contained in the surface of the Face A.
11.4 Results Model IV
The results of Model IV are an analytical implementation of the expressions presented in Section9.1.2. By comparing the results for Kaolin Clay in Fig. 28 with the Model II COMSOL resultsalso for Kaolin Clay (Fig. 22, black dashed line) it is obtained the same result for lateral earthstress. That is to say, Models II and IV are verified.
11.5 Results Model VI
The effect of the Fluid and Seabed interactions is shown in the model results. Firstly, Fig. 29 showthe 3 principal stresses distribution and direction along the structure surface. The 1st principalstress plot (a) shows a stress concentration on the Machine Hall ceiling pointing downwards (neg-ative values of the stress) and shear stress at the shaft. As for the 2nd principal stress, it accountsfor shear stresses on the Machine Hall ceiling, and downwards stresses on the shaft and on theMachine Hall ceiling. Finaly, plot (c) accounts for the shear stress of the surfaces studies.
Additionally, Von Mises stress is plotted in Fig. 30. It is show again that the larges stresses arelocated on the Machine Hall ceiling and lateral surfaces. The central linear surface of the shaft alsohave significant stress. Finally, Von Mises stress accumulated in the interior edges of the machinesHall.
The displacement result on Fig. 23 show the consequences of the stress concentration on theMachine Hall ceiling observed before: there is a maximum downwards displacement of 3.61mm ofthe centre of the structure. Additionally, a small tilt of approximately 2.5mm is observed on thetop of the shaft.
57
Lateral Earth Pressure at-rest (N/m2)
water level
seabed surface
Depth below
water level (m)
Depth below
seabed surfacedseabed
0 1.83e5 0 1.93e5 4.68e5 0 1.93e5 6.51e5
σr,waterσr,soil
σr
z(m)
+ =
Figure 28: Lateral earth pressure at-rest as a function of depth below the seabed surface (σr(z)).Used dseabed = 35m and soil parameters of Kaolin Clay (Table 2).
58
(a) First Principal Stress (N/m2) (b) Second Principal Stress (N/m2)
(c) Third Principal Stress (N/m2)
Figure 29: First, second and third principal stresses on the structure external surface due to one-way interaction of fluid and seabed to the Engine Reservoir. Stress magnitudes as surface plot,stress directions as arrows.
59
(a) Walls External Surface (b) Walls Internal Surface
Figure 30: Von Mises stress (N/m2) for the external (a) and internal (b) wall surfaces of the EngineReservoir structure.
Figure 31: Total displacement magnitude (m) and scaled deformation of the Engine Reservoirexternal wall surface using pure concrete walls.
60
11.6 Results Model VII
Model VII generates a enormous quantity of results, as it details the reinforcement for every struc-tural component of the structure. It has only been added the more relevant results for the resultsanalysis, but the rest of results are accessible through CYPECAD model.
Total displacement is shown in Fig. 32. It is again observed a maximum displacement on themiddle of the Machine Hall of 2.97mm. It differs from the Model VI maximum displacement for0.66mm. Nevertheless, the displacement on the shaft top is of order of magnitude O(-1) mm.Figure 33(a) shows the z-displacement of the Machine Hall; its maximum value at the centre is-2.97mm and corresponds with the maximum total displacement found previously. Therefore, it isclear to see that displacement z clearly predominates over x and y displacements. The momentumdistributions on subplots (b) and (c) show clearly the effect of the ceiling bending downward fromthe centre. Figure 34 shows the desired phenomena of the columns absorbing the shear stress, boththe x-direction and y-direction shear stresses.
Additionally, subplots from Fig 35 represent the (averaged) ratio of needed reinforcement. Asobservable, the recommended reinforcement distribution for the inferior and the superior rein-forcement (with respect to slab centre) greatly differs, as well as for the inferior x and y-directionreinforcement.
Finally, the results for the reinforcement calculations are presented in Fig. 36. The cross at themiddle of the floor means there is a base reinforcement, and the additional lines (as in (b)) specifythe additional reinforcement that has been considered suitable for the loads applied. P2 and P5slabs have only base reinforcement, but P3, which refers to the Machine Hall ceiling, has a verysignificant ratio of reinforcement, with a base grid of steel bars. A more clear data is obtainedwhen the quantities of material used are evaluated and presented in Tab. 12 and Fig. 36(a,b). Avery large amount of material is required for the project construction: 59357 kg of steel bars and671 m3 of concrete. Furthermore, Fig36(a) shows P4 is the floor with highest material domainfor both concrete and steel, as it contains most of the Engine Reservoir Shaft. It is also worthmentioning the high requirement of steel in P3, which refers to the dense reinforcement of theMachine Hall ceiling.
Figure 32: 3D representation of the deformation suffered on the Engine Reservoir walls, slabs andpillars.
61
(a) Displacement, z-direction (mm)
(b) Moment, x-direction (t·m/m) (c) Moment, y-direction (t·m/m)
Figure 33: Displacement in z-direction and moments in x,y-direction on the upper concrete slabof the Machines Hall.
Table 12: Project summary in terms of concrete surface (m2), concrete volume (m3) and reinforce-ment steel bars mass (kg).
62
(a) Total shear stress (t/m)
(b) Shear stress, x-direction (t/m) (c) Shear stress, y-direction (t/m)
Figure 34: Total and x,y-direction shear stresses on the upper concrete slab of the Machines Hall.
63
(a) Inferior, x-direction (b) Inferior, y-direction
(c) Superior, x-direction (d) Superior, y-direction
Figure 35: Amount of reinforcement steel required (cm2/m) for the upper concrete slab of the Ma-chines Hall. Reinforcement considered: superior and inferior reinforcement for both x,y-directions.
64
(a) Floor P2 (b) Floor P3 (c) Floor P4 (d) Floor P5
Figure 36: Slab Reinforcement of the project floors P2-P5. Floor P1 is not added because itrepresents the foundations, it does not contain an horizontal slab.
(a) (b)
Figure 37: Decomposition of the steel mass (a) and the concrete volume (b) used in each floor andfor each structural component.
65
12 Discussion
Section 7 presents a detailed methodology for designing some of the components of the EngineReservoir: the reversible pump/turbine main characteristics, the penstock diameter, the reservoirscapacity and the pump flow rate. It has been observed the relation with the component designvariable with the turbine design head, which stresses the relevance of a good design of the turbinedesign head. Nevertheless, the hydro-power and electrical components characterisation is not com-plete due to difficulties in finding detailed datasheets of the components.
Once the main components have been characterised, the designs are developed in Section 8. DesignI was made before the completion of the hydro-power components characterisation, therefore it doesnot present a clear distribution or design of the components inside the Machine Hall. Neverthe-less, it presents the advantages of more light simulations (highly symmetric) and greater resistanceagainst hydrostatic pressures and lateral earth pressures. As a contrast, Design II presents a de-tailed design of the Machine Hall for the elements that could be characterised.
Section 6.0.1 describes in detail the main interactions to be modelled and tested in a ordered way.Both the stability conditions, and the seabed and fluid interactions are clearly presented and readyto be implemented in the models. On the other side, the interactions with the Rigid and FlexibleReservoirs are treated in less detailed, and its effects quickly neglected. Although the approximatedcalculations show they do not cause any significant load that could make the Engine Reservoir looseits stability, there could be significant concentrated loads and momentum on certain regions of thestructure, specially on the flow connections with the reservoirs. It could be argued that a depthstudy on Engine Reservoir connections would be necessary before neglecting the interactions.
Regarding the interactions and structure modelling, Model I provides a clear insight on structurestability on a porous seabed, and provides a useful design criteria (safety factors satisfaction) anddesign requirement (specified safety factor) to be taken into account in next steps of the designproject or in other design projects. Model II and IV verify the expressions from Section 39 regard-ing the pore pressure, effective vertical pressure and lateral earth stress. Additionally, Model IIprovides results for soil compression, although it is of few interest for the current design. ModelIII proves the currents can be discarded at the range of seabed depth of the current design, wherehydrostatic pressure is orders of magnitude bigger than the viscous forces caused by the currents.Nevertheless, the data and fitting procedures from Appendix A,B,C might be used for other projectthat contain solid domains near or partially above the fluid surface.
Model V was a very promising idea, as it took advantage of the COMSOL’s ability to combinePhysics to unify all the previous models. A mistake on the governing equations or boundary condi-tions must have been made. As for Model VI, it satisfactorily models the environment interactionswith the Design II structure. It is observed a accumulation of error on the ceiling of the MachinesHall and a small tilt along the structure shaft. It is concluded that steel reinforcement is necessaryfor reducing the bending displacement of the concrete material. On the other side, additionaloptimisations could be made on the structure geometry at this stages, so that the concrete receivescompressing stresses. Design I geometry could be recovered at this point, or a dome-based shapeof the Machines Hall geometry might have a better performance.
Finally, Model VII provides detailed results on the structure components and building drawings.Small moments are observed, and the shaft suffers a significantly lower displacement than in thecase of no reinforcement. Moreover, the slabs and walls displacements when compared to the no-reinforced model have also decreased. Nevertheless, the resulting design needs too many, it is noteconomically feasible. It is expected that the geometry optimisation mentioned in the previousparagraph will greatly reduce the high reinforcement requirement of the design.
66
13 Conclusion
Several goals were defined for the design project. Its evaluation will determine the success of thedesign project.
The first goal was to identify the Engine Reservoir design requirement. They have been clearlyidentified in Table 1 and also related to design criteria. Therefore, the first goal has been achieved.
The next goal was to make a preliminary design of the hydropower components. Following theargumentation in the Discussion section, it is though that the goal has been half achieved. On theone side, some components have been described in detail, such as the reversible pump-turbine andthe penstock. On the contrary, other components such as the motor and the generator have beencharacterised using a lot of approximations due to the lack of available data on the flied. Others,such as the transformers, have not been treated. Conclusively, this goal is half achieved.
The third and forth goal were to identify the main interactions between the Engine Reservoir andthe fluid, the seabed and the Rigid and Flexible Reservoirs, and to develop a method for the in-teractions quantification. According to what has been said in Discussion section, the interactionsbetween the Engine Reservoir and the environment are identified and described in detail, andthe main governing equations of the interactions are also provided. Additionally, multiple modelshave been developed that study and simulate the interactions individually, so that the effect ofdesign variables such as the embedded depth on the studied interaction are more easily detectedand understood. Nevertheless, the interaction between Engine Reservoir and Flexible and RigidReservoir should be treated in more detailed. Specifically for the connection component betweenthe two structures, it should be analysed and tested individually. As a result, the goal is consideredachieved for the most part.
Finally, the fifth goal was to elaborate a first design for the Engine Reservoir and to analyse itsmechanical performance. Design II has been developed taking into account all the requirementsfrom Tab. 1 and from the hydro-power machines characterisation. Not only the external geometryhas been defined, but also the general distribution of the components within the Machine Hall (Fig.10) and the main structural components (Fig. 17). Additionally, an extensive mechanical analysishas been performed with results regarding the structure displacement, momentum, stress distribu-tion, reinforcement ratio requirement an others (Sec. 11.6). Finally, it is also provided a projectsummary regarding the materials in total and decomposed in floors and structural components. Ithas been concluded that the model was not economically feasible due to the large amount of steelused, but the model is very flexible for changes on the structure geometry or material. Then, it isconcluded that the fifth goal is reached satisfactorily.
67
14 Recommendations and Limitations
Along the project, several limitations of the models and the design has been detected. Some ofthem are caused by lack of information (such as seabed characteristics or machines characteristics)and some of them for the sake of simplifications in the models (one-way interactions). All of themwill be explained below, and some recommendations will be made regarding model and designfurther modifications.
The lack of knowledge on the seabed characteristic is a big limitation of the design. Along theproject and specially on the models results, it has been observed a big dependency of the soil typefor the uplift stability factor, the lateral earth pressure calculation and the filling soil weight. Theseparameters have a high effect on the structure design, as for example large lateral earth pressureswill require a more robust structure. Therefore, the more information of the soil is provided, themore can the design be optimised.
Also in the field of seabed characterisation, no different soil layers have been treated during theproject. One consequence of considering multiple soil layers (of different density and earth pressurecoefficient value) would be a change on the earth pressures on the structure boundaries. Anothereven more drastic change on the model would be the consideration of impermeable layers, whichwould cause the water pore pressure to suddenly drop, and therefore the earth pressures on thestructure surfaces would change drastically. Additionally, all models assume completely saturatedsoil at all soil depth, although partially saturated soils can also be found.
As for the hydro-power machines characterisation, there is a lot of work left on characterising theparameters. Moreover, the design choice of 4 turbines of 500kW each should be reviewed by takinginto account the turbine efficiency with respect to the head, the head variation, the designed headan other turbine parameters that have to be specified in detail to offer the maximum efficiency. Thesame recommendation applies for the pump and other relevant components of the Engine Reservoir.
A big limitation of the project design is that is has not taken into account the installation andconstruction processes for the structure design. Literature research should be done on the em-bedding procedure procedures of large structures. Alternatively, the design in parts of the EngineReservoir could be considered. Also the transportation process from the shore to offshore andto the bottom of the lake is to be investigated. In conclusion, construction, transportation andinstallation procedures should be detailed, and the design of the Engine Reservoir might changeto adequate to the procedures requirements.
Finally, an economic study should be made for the design of the structure. Specifically, the em-bedded depth plays an important on machinery characterisation, power production and storagecapacity, minimum strength of the structure and so on. The materials summary on Tab. 12 is thefirst approach to include economical considerations into the design.
68
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windturbine-wes-250/#toggle-id-6. [Accessed in 28-10-2020].
[32] Anson Steel. Jis g3101 ss400 structural carbon steel platespecification. http://www.steels-supplier.com/steel-standard/
jis-g3101-ss400-structural-carbon-steel-plate-specification.html. [Accessedin 12-11-2020].
[33] Sulzer. Pumped hydro storage power. https://www.sulzer.com/-/media/files/
applications/power-generation/brochures/pumped_hydro_sstorage_power_e10125.
ashx?la=en. [Accessed in 15-10-2020].
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A Metocean Data
Metocean data refers to data related with meteorology and (physical) oceanography. The meteo-cean conditions of the design project are the combination of wind, current and wave conditionsof the chosen location. The data is categorised as normal sea-state conditions (Tab. 13 and 14)and extreme sea-state conditions (Tab. 15). More thorough and detailed meteocean studies willbe required in further phases of the design (out of scope).
Metocean data is extracted from the database generated by Rijksdienst voor Ondernemend Neder-land (RVO.nl). The report produced in 2019 called ’Metocean desk study and database for DutchWind Farm Zones’ gathers metocean data from 15/1/1979 to 1/10/2018 and its pre-processedanalysis [5]. The dataset has been published by DHI as ’MIKE 21 Hydrodynamic Model (HD)’,and it covers the Dutch Offshore Wind Farms area delimited in Fig. 38. In particular, the tablespresented bellow contain the data from the area of Ten Noorden van de Waddeneilanden (TNW)offshore wind farm (associated with number 5 in the figure).
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Figure 38: Dataset covered area (delimited with a purple line) and colour-map of the mean waterdepth (land areas in grey). Included Dutch offshore wind farms (delimited by a black line): Borssele(1), Hollandse Kust zuid (2), Hollandse Kust noord (3), Hollandse Kust west (4), Ten Noordenvan de Waddeneilanden (5) and IJmuiden Ver (6) [5].
Depth Parameter Mean [m/s] Min [m/s] Max [m/s]
Depth-integratedutot,depth−int 0.3 0.0 1.0utid,depth−int 0.3 0.0 0.6ures,depth−int 0.1 0.0 1.0
Near-surfaceutot,surf 0.4 0.0 1.3utid,surf 0.3 0.0 0.8ures,surf 0.1 0.0 1.4
75% of water columnutot,75% 0.3 0.0 1.1utid,75% 0.3 0.0 0.7ures,75% 0.1 0.0 1.2
50% of water columnutot,50% 0.3 0.0 1.1utid,50% 0.3 0.0 0.6ures,50% 0.1 0.0 1.0
25% of water columnutot,25% 0.3 0.0 0.9utid,25% 0.2 0.0 0.5ures,25% 0.1 0.0 0.8
5% of water columnutot,5% 0.1 0.0 0.5utid,5% 0.1 0.0 0.3ures,5% 0.0 0.0 0.5
Table 13: Current velocity data for TNW normal sea state (’Mean’) and extreme sea state (’Max’).Current depth average and current velocity profile data for specific percentages of the water column[5].
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HS,NSS [m]TP,NSS [s]
5% 50% 95%1 3.4 5.4 14.6
1.5 4.4 6.0 12.42 5.2 6.5 12.1
2.5 5.9 7.1 11.13 6.6 7.7 10.9
3.5 7.3 8.2 10.24 7.9 8.6 10.4
4.5 8.4 9.1 11.05 8.9 9.6 11.4
5.5 9.3 10.1 12.36 9.7 10.5 12.3
6.5 10.2 11.1 13.07 10.5 11.6 14.1
7.5 11.0 12.5 14.08 11.2 13.0 15.0
Table 14: Significant wave height and associated peak wave period for TNW normal sea state. Thesignificant wave height (HS,NSS) is discretised by 0.5m steps, and the range of peak wave period(TP,NSS) associated to each wave height is presented by the values of 5%, 50% and 95% of thedata [5].
Parameter Symbol Value UnitExtreme total high water level HWLm 3.5 mExtrem total low water level LWLm -1.0 mExtreme wind speed(10m from MSL and 10min averaged)
um,10m,10min 34.4 m/s
Significant wave height Hs 9.1 m/sMaximum wave height Hm 17.2 m/sMean wave period associated to maximum wave height THm
11.8 s
Table 15: TNW metocean extreme values with a 50-years return period [2].
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B Hydrodynamic environmental modelling
The design has to take into consideration the environmental conditions that have a bigger impacton its structural behaviour: the currents and the waves. From measurements or model-generateddata, two environmental conditions have to be described:
• Normal environmental conditions: it is expected that the normal environmental conditionsoccur frequently during the service life of the Engine Reservoir.
• Extreme environmental conditions: it is expected that the extreme environmental conditionsoccur rarely during the Engine Reservoir service life. They constitute the maximum loadingof the Engine Reservoir.
The structure is exposed to the sea state currents and waves. The principles that guide this actionsand its interaction with the half-buried structures are presented below and they are based on thework made by Kokkinowrachos on ´Hydrodynamik der Seebauwerke´.
B.0.1 Currents modelling
The currents present on the sea are commonly categorised into tidal currents (associated with as-tronomical tides), wind-generated currents and, if applicable, wave-induced currents and speciallybreaking waves.Depending on the location, other currents may be considered, such as the permanent current cur-rent (constant velocity over the depth).
The current velocity vector has a spacial and temporal dependency, it can be generally written asuc = uc(x, y, z; t). The time-dependency that originates flow fluctuations is caused by turbulence.Nevertheless, for most applications the current can be considered as a steady flow, where the cur-rent velocity only depends on the depth (vc = vc(z)).
The total current velocity at a given location (x,y) is the vector sum of the each type of currentpresent, and the total current profile describes its speed and direction as a function of the elevation.It should be described for both normal and extreme environmental conditions.
Tidal currentTidal currents are associated with astronomical tides, they are caused by the gravitational forcesof the moon and the sun. They are generally weak on deep waters past the shelf break, andstronger on broad continental shelves. Its normal current values rarely exceed 0.3 m/s along anopen coastline When there are no detailed field measurements, the tidal current velocity vc,tidalmay be modelled by the following power law equation assuming unidirectional current:
uc,tidal(z) = uc,tidal(0) ·(d+ z
d
)αfor z ≤ 0 (107)
where z is the water level, being z = 0 the still water level, uc,tidal(0) the tidal current velocity atstill water level, d the seabed depth and the exponent is taken generally as α = 1/7.
Wind-generated currentThe wind-generated current velocity vc,wind may be modelled as a linear profile from z = d0 tostill water level:
uc,wind(z) = f(x) =
{uc,wind(0) · d0+zd0
for −d0 ≤ z ≤ 0
0 for z < −d0(108)
where the reference depth is generally chosen as d0 = min (dmudline, 50m) and the wind-generatedcurrent at still water level is given by uc,wind(0) = 0.015 · v10m,1h, with u10m,1h being the averageover 1h of the wind speed at 10m height above the still water level. Conversely, the wind-generatedcurrent from z = d0 to still water level can be taken as a slab profile:
uc,wind(z) = f(x) =
{uc,wind(0) for −d0 ≤ z ≤ 00 for z < −d0
(109)
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From the two expressions of the wind-generated current profile, it has to be applied the profilethat gives the highest loads for the Engine Reservoir design.
Wave-generated currentThe wave-generated currents, and in particular the surf-generated currents, are to be consideredwhen breaking waves are present on the water surface. The surf-generated current is estimablethrough numerical models, such as the Boussinesq model. For coastal currents parallel with theline of the coast, the design velocity can be approximated as:
uc,surf = 2 · s ·√g ·HB , (110)
where s is the seabed slope near the shore, and HB is the height of the breaking wave. Thewave-generated current velocities can be determined by the same power law equation as the tidalcurrents:
uc,surf (z) = uc,surf (0) ·(d+ z
d
)1/7
for z ≤ 0, (111)
Superposition of currentsThe flow velocity that determines the velocity vertical profile of the sea-water is obtained by thecombined action of the previously defined sea currents. For simplicity and safety, the currents areassumed in the same direction, and therefore the total current velocity uc(z) as a function of thevertical component is obtained by:
uc(z) = Uc,tidal(z) + uc,wind(z) + uc,surf (112)
B.0.2 Waves modelling
For the wave model, different models should be applied depending on the mean water depth, thewave height and the wave period. The regions of applicability of the main wave theories are shownin Fig. 39. The figure shows their dependency with two dimensionless wave properties: the di-mensionless wave steepness H and the dimensionless relative depth d:
H =H
gT 2, d =
d
gT 2(113)
where H is the wave height (m), T is the wave period (s), g is the gravity acceleration (m/s2) andd is the mean water depth or MWD (m).
Firstly, the applicable wave theory is evaluated for the normal sea state case. Using Eq. 113 andmean water depth d = 35m, the metocean data for normal sea state conditions (H = HS,NSS and
T = TP,NSS from Tab. 14) is used to computeH and d. They are compared with the regionsof applicability of Fig. 39, and it is obtained the applicable wave theory for every conditionconsidered. The results are summarised in Tab. 16, from which it is concluded that he waves ofthe normal sea state should be modelled using Stokes 5th order or Stream Function.Secondly, the extreme case is considered. From Tab. 15, the extreme wave parameters are H =Hm = 17.2m/s, T = THm = 11.8s and dHWL = d+HWLm = 38.5m and dLWLL = d+WHLm =34m. Substituting the extrem values in Eq. 113, it is obtained Hm = 0.0126 and dm,HWL = 0.028
and dm,HWL = 0.025 for the case of extreme total high wave level and low wave level respectively.Comparing the results with the regions of applicativity of Fig. 39, it is obtained that the fluidparticle motion may be computed using Stokes 5th order and Stream Function 3rd order. are alsoappropriate for the extreme sea state.Therefore, the decision on the wave model to use for the design wave loads modelling is dependenton the location and wave data.
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H [m] T [s] H d Region of applicability
1 3.4 0.0088 0.309 Stokes 5th order / Stream Function 3rd order1 5.4 0.0035 0.122 Stokes 5th order / Stream Function 3rd order1 14.6 0.0005 0.017 Stokes 5th order / Stream Function 3rd order
4.5 8.4 0.0065 0.051 Stokes 5th order / Stream Function 3rd order4.5 9.1 0.0055 0.043 Stokes 5th order / Stream Function 3rd order4.5 11.0 0.0038 0.029 Stokes 5th order / Stream Function 3rd order (limit)8 11.2 0.0065 0.028 Stokes 5th order / Stream Function 3rd order8 13.0 0.0048 0.021 Stokes 5th order / Stream Function 3rd order8 15.0 0.0036 0.016 Stokes 5th order / Stream Function 3rd order (limit)
Table 16: Computation of dimensionless wave steepness (H) and relative depth (d) for the caseof normal sea states. Results for the lowest, intermediate and largest significant wave height andtheir associated peak wave periods (Tab. 14).
Figure 39: Regions of applicability of different wave theories as a function of the dimensionless wavesteepness (H/gT 2
app) and the dimensionless relative depth (d/gT 2). The wave theories consideredare Stream Function, Stokes V and Linear Wave Theory [1]
76
C Velocity profile fitting for normal and extreme state con-ditions
From normal and extreme sea states, current input data (Tab. 13 have been used to fit nonlinearmodels. It is direct to see from the tables that the current is mainly caused by tidal currents; thisobservation will influence in the proposed nonlinear models for the current profile. Additionally,it has been imposed null current velocity at mudline depth to satisfy the boundary conditionsbetween fluid and seabed sub-models (referred in the figure as ’mudline condition’).
The nonlinear models expressions have been extracted from Section B.0.1. In both cases (normaland extreme sea states), two different procedures are used to fit the data. Nonlinear model #1is constructed by firstly fitting the tidal current data (utid) from Tab. 13 to the theoretical tidalcurrent expression (Eq. 114), and secondly using the total current data to fit the total currentmodel (Eq. 115), where c1, α have been already determined in the first fitting procedure. Then,the model #1 is characterised by coefficients c1, α and c2 as:
utidal(z) = c1
(d+ z
d
)α(114)
utotal(z) = utidal(z) + c2
(d+ z
d
)(115)
Conversely, the nonlinear model #2 is directly defined as:
utotal(z) = c1
(d+ z
d
)α, (116)
where coefficients c1 and α have to be fitted.
The results of the fitting are shown in Fig. 40, and summarised in Tab. 17,18. From the comparisonof the Root Mean Square Error and Adjusted R-Squared Error, it has been chosen Model #1 formean current velocity expression, and Model #2 for the case of maximum current velocity. Then,the velocity profile for normal sea-state is calculated using Eq. 117 (unss(z)), and extreme sea-stateis obtained from Eq. 118 (uess(z)):
unss(z) = 0.33(35 + z
35
)0.34+ 0.06
(35 + z
35
)(117)
uess(z) = 1.28(35 + z
35
)0.29(118)
(119)
Model #1 Model #2Fitted coefficients c1 = 0.33, α = 0.34, c2 = 0.06 c1 = 0.38, α = 0.34Root Mean Square Error 0.0424 0.0439Adjusted R-Squared Error 0.921 0.915
Table 17: Fitted coefficients and fitting errors of the nonlinear models regarding the mean currentvelocity.
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(a) Mean current velocity (b) Maximum current velocity
Figure 40: Measured current velocity profiles (orange circles, data from Tab. 13) and predictedvelocity profiles (continue curves). The velocity profiles have been predicted using the nonlinearmodel from Eq. 114,115 (blue) and from Eq. 116 (red)
Model #1 Model #2Fitted coefficients c1 = 0.78, α = 0.33, c2 = 0.62 c1 = 1.28, α = 0.29Root Mean Square Error 0.166 0.056Adjusted R-Squared Error 0.882 0.987
Table 18: Fitted coefficients and fitting errors of the nonlinear models regarding the maximumcurrent velocity.
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